Studies on a Class of Vertical, Submerged and
Floating Breakwaters in a Two-Layer Fluid
A thesis submitted to the
Indian Institute of Technology, Kharagpur
for the award of the degree of
Doctor of Philosophy
By
P. Suresh Kumar
Department of Ocean Engineering and Naval Architecture
Indian Institute of Technology
Kharagpur - 721 302, INDIA
2007
Studies on a Class of Vertical, Submerged and
Floating Breakwaters in a Two-Layer Fluid
A thesis submitted to the
Indian Institute of Technology, Kharagpur
for the award of the degree of
Doctor of Philosophy
By
P. Suresh Kumar
Department of Ocean Engineering and Naval Architecture
Indian Institute of Technology
Kharagpur - 721 302, INDIA
2007
Dedicated to
Dr. T. D. Singh (H.H. Bhaktisvarupa Damodara Swami)
my Spiritual Master, for his Encouragements for the Research Work
and
Supreme Personality of Godhead, Lord Jagannath
for His Unlimited Love and Blessings
CERTIFICATE
This is to certify that the thesis entitled "STUDIES ON A CLASS OF VERTICAL,
SUBMERGED AND FLOATING BREAKWATERS IN A TWO-LAYER
FLUID" being submitted by Mr. P. Suresh Kumar to the Indian Institute of Tech-
nology, Kharagpur, for the award of the degree of Doctor of Philosophy, is a record of
bonafide research work carried out by him under our supervision and guidance and that
Mr. P. Suresh Kumar fulfills the requirement of the regulation of the degree. The re-
sults embodied in this thesis have not been submitted to any other University or Institute
for the award of any degree or diploma.
Trilochan Sahoo
Associate Professor
Dept. of Ocean Engng. &: Naval Arch.
Indian Institute of Technology,
Kharagpur — 721 302, India
Debabrata Sen
Head of the Department
Dept. of Ocean Engng. h Naval Arch.
Indian Institute of Technology,
Kharagpur — 721 302, India
Acknowledgments
After the completion of this thesis, I am experiencing sincere feelings of achievement
and satisfaction. Looking into the past I realize how impossible it is for me to succeed
on my own. I wish to express my deep gratitude to all those who have extended their
helping hands towards me in various ways during my short tenure at Indian Institute of
Technology, Kharagpur.
It gives me immerse pleasure to express my deep sense of gratitude and heartily
thanks to my supervisors Professor Trilochan Sahoo, Department of Ocean Engineer-
ing and Naval Architecture, Indian Institute of Technology, Kharagpur and Professor
Debabrata Sen, Head of the Department, Ocean Engineering and Naval Architecture,
Indian Institute of Technology, Kharagpur. I am thankful for their invaluable guidance
and constant encouragement.
I am highly indebted to Dr. S.R. Manam, Postdoctoral Fellow, Civil and Environ-
mental Engineering, Technion, Haifa Israel, for his help towards formulating the research
problems in my Ph.D. work. I am very much inspired by his continuous encouragements
and guidance during my Ph.D. work.
I would like to express my sincere thanks to my DSC members Professor N.R.
Mandal, Department of Ocean Engineering and Naval Architecture, Indian Institute
of Technology, Kharagpur and Professor S.K. Satsangi, Dean (P G and R), Indian
Institute of Technology, Kharagpur, for their words of wisdom, which helped me to remain
focused in the research work at a very difficult phase of my Ph.D.
I sincerely acknowledge other faculty members of my department. Professor A.
Bhar, Professor S.C. Misra, Professor O.P. Sha, Professor A.H. Sheikh, Profes-
sor H.V. Warrior, Professor A.K. Otta and Professor P.K. Bhaskaran for their
valuable suggestions.
I am grateful to Mr. P.K. Ray for his constant help during the entire period of my
Ph.D. My sincere thanks and gratitude are due to my colleagues. By being with them I feel
myself very fortunate: Dr. P. Topdar, Dr. M. Adak, Dr. S. Chakraborty, Dr. A.
Chakraborty, Mr. R. Sharma, Dr. J. Bhattacharjee, Mr. C. Padhy, Mr. M.C.
Manna, Dr. R. Datta, Mr. S. Das, Mr. A. Bhar, Mr. R. Kumar, Mr. S. Maity,
Mr. A. Tyagi, Mr. P. Biswas, Ms. S.S. Phoenix, Ms. B. Chakraborty, Mr.
M.K. Pandit, Mr. D. Karmakar, Mr. A. Datta and Mr. S.P. Singh. Thanks are
also due to the other staff members of the Department who have always been cooperative
and have been a constant source of encouragement to me. I sincerely acknowledge Dr.
Y.M. Oh, Principal Researcher, Coastal Engineering Research Department, Korea Ocean
Research and Development Institute, Korea, for his support in printing the final version
of my Ph.D. thesis.
I take this opportunity to express my sincere thanks to my friends, Mr. P. Banerjee,
Professor S. Ghosh, Mr. D. Khan, Mr. G. Silwal, and Mr. S.N. Patel for their
help in crucial moments over the entire period of the work. I have many more friends who
helped and encouraged me sincerely. I thank them all from my heart.
I offer my prayers to my spiritual masters Dr. T. D. Singh (H.H. Bhaktisvarupa
Damodara Swami) and H.H. Bhakti Madhava Puri Swami who have always in-
spired me to look into the deeper aspects of science and their words of wisdom is a great
asset. I also offer my sincere prayers to Supreme Personahty of God Head, Lord
Jagannath for His blessings and mercy upon me during the entire period of my Ph.D.
Lastly I am grateful to my family members who have spared me from the responsibil-
ities and provided me the moral support and encouragement to carry out the entire work
successfully.
Dated: (P. Suresh Kumar)
Preface
In this thesis, a class of problems pertaining to scattering and trapping of harmonic
surface- and internal-waves by various types of breakwaters, namely (i) Rigid dikes, (ii)
Porous membrane and (iii) Flexible porous plate in a two-layer fluid are studied. These
physical problems, under the assumption of the linearized-theory, are reduced to a class
of two-dimensional mixed boundary value problems which are then solved for the un-
known velocity potentials along with important physical quantities like the reflection and
transmission coefficients of an incident time-harmonic wave. Assuming small amplitude
response in the cases of flexible structures such as porous membrane and porous plate,
the general structural response equation which is coupled with the velocity potentials has
been utilized to determine the structural response.
Standard mathematical techniques are utilized, in the reduction and solution of the
boundary value problems, such as eigenfunction-expansion method, wide-spacing-approxi-
mation method (WSAM) and the least-squares-approximation method.
The content of the thesis is presented in the form of nine chapters. Chapter 1 is
devoted to a general introduction and the objectives of the present study. In Chapter
2, an elaborate review of literature and the motivation for the present investigation are
presented. The basic mathematical tools utilized in the thesis along with the derivations
of the basic hydrodynamics and structural response equations in the linearized set up
are elaborated in Chapter 3. Although these are available in various text books in a
scattered manner, the purpose of this chapter is to present the underlying mathematical
formulation in a coherent and connected manner so as to make the thesis self-sufficient.
In Chapter 4, surface- and internal-waves scattering by a single surface-piercing rigid
dike is investigated numerically within the context of linearized-theory of water waves.
After solving this physical problem the study is extended to a pair of identical rectangular
surface-piercing dikes. The surface- and internal-waves scattering by a single bottom-
standing rectangular rigid dike and a pair of bottom-standing rigid dikes are presented in
Chapter 5. In these works, the geometrical symmetry of the problems is being exploited
by splitting the velocity potentials into symmetric and antisymmetric components. The
solution of the boundary value problem is derived by matched-eigenfunction-expansion
method. Because of the flow discontinuity at the interface, the eigenfunctions involved
have an integrable singularity at the interface and the orthonormal relation used in the
present analysis is a generalization of the classical one corresponding to a single-layer fluid.
Computed results in two-layer fluid are compared with those existing in the literature
for a single-layer fluid. Moreover, the results obtained by the matched-eigenfunction-
expansion method are compared with that of WSAM. The wave reflection characteristics
of the system subject to normal incident waves are investigated. The force amplitudes
are computed and analyzed for various physical parameters.
In Chapter 6 and Chapter 7, surface- and internal-waves scattering by flexible
porous structures is considered. Mathematical models are developed to solve the complex
physical problems such as flow past porous flexible structures in a two-layer fluid. In
Chapter 6, the scattering of water waves by a flexible porous membrane breakwater in
a two-layer fluid having a free surface is analyzed in two-dimensions. Linear wave theory
and small amplitude membrane response is assumed. The porous-effect parameter used
in the study is a complex number, which includes both inertia and resistance effects.
The porous membrane breakwater is tensioned and pinned at both the free surface and
the seabed. The associated mixed boundary value problem is reduced to a linear system
of equations by utilizing a more general orthogonal relation along with least-squares-
approximation method. The reflection and transmission coefficients for the surface- and
internal-waves, free surface and interface elevations, and the non-dimensional membrane
deflection are computed for various physical parameters like the non-dimensional tension
parameter, porous-effect parameter, ffuid density ratio, ratio of water depths of the two
fluids, to analyze the efficiency of a porous membrane as a wave barrier in the two-layer
fluid. The problem is then extended to the case of a flexible plate in Chapter 7. The
plate breakwater is extended over the entire water depth and the problem is analyzed in
two-dimensions. The reflection and transmission coefficients for the surface- and internal
modes, wave load and breakwater response are computed for various physical parameters
of interest to analyze the efficiency of the flexible porous plate as a breakwater in the
two-layer fluid.
In Chapter 8 the surface- and internal-wave trapping by porous and flexible partial
breakwaters near the end of a semi-inflnitely long channel is studied in two-dimension.
In the study, both surface-piercing and bottom-standing conflgurations are considered.
The surface-piercing breakwater is clamped above the free surface and is free at the other
end which is submerged in the fluid. On the other hand, the bottom-standing break-
water is flxed at the seabed and the other end is having a free edge. A combination of
eigenfunction-expansion method and least-squares-approximation method is used to solve
the associated mixed boundary value problems. The reflection coefficients are obtained
and discussed for both surface- and internal-waves for different values of non-dimensional
ffuid density ratio, porous-effect parameter, normalized distance between breakwater and
channel end-wall, length of submergence and ffexural rigidity of the breakwaters for both
surface-piercing and bottom-standing cases. Furthermore, the hydrodynamic force act-
ing on the ffexible partial breakwaters and the breakwater response are determined for
different physical parameters of interest.
Chapter 9 provides concluding remarks on the study. The scope of the present
research work is also described followed by future scope of work. Bibliography has been
included separately.
Keywords:
Surface-waves; Internal-waves; Wave reflection; Wave transmission; Wave scattering;
Wave trapping; Resonance; Dissipation; Dikes; Flexible breakwaters; Porous breakwa-
ters; Surface-piercing breakwaters; Bottom- standing breakwaters; Eigenfunction-expansion
method; Least-squares-approximation; Wide-spacing-approximation; Orthogonal relation;
Contents
List of Figures vii
List of Notations xiii
1 INTRODUCTION 1
1.1 TWO-LAYER FLUID 1
1.2 WAVE-STRUCTURE INTERACTION 4
1.3 BREAKWATERS 6
1.3.1 Rigid Breakwaters 7
1.3.2 Flexible Breakwaters 8
1.3.3 Porous Breakwaters 9
1.3.4 Partial Breakwaters 10
1.4 OBJECTIVE OF PRESENT INVESTIGATION 12
2 REVIEW OF LITERATURE 15
2.1 INTRODUCTION 15
2.2 WAVE-STRUCTURE INTERACTION IN A SINGLE-LAYER FLUID . . 16
2.2.1 Rigid Breakwaters 17
2.2.2 Flexible Breakwaters 24
2.2.3 Porous Breakwaters 32
2.3 WAVE-STRUCTURE INTERACTION IN A TWO-LAYER FLUID .... 39
2.4 MOTIVATION FOR THE PRESENT INVESTIGATION 42
i
ii CONTENTS
3 GENERAL MATHEMATICAL FORMULATION 43
3.1 INTRODUCTION 43
3.2 MATHEMATICAL MODEL FOR TWO-LAYER FLUID 44
3.2.1 Definition of Velocity Potential and Governing Equation 45
3.2.2 Linearized Free Surface Boundary Conditions 46
3.2.3 Linearized Interface Boundary Conditions 47
3.2.4 Boundary Condition on the Rigid Boundaries 48
3.2.5 Radiation Conditions at Infinity 48
3.2.6 Continuity Conditions Across the Gap 49
3.3 MATHEMATICAL MODEL FOR BREAKWATER RESPONSE 50
3.3.1 Governing Equation 50
3.3.2 Edge Conditions 51
3.3.3 Continuity Condition Across the Free Surface 52
3.3.4 Continuity Condition Across the Interface 52
3.4 CONDITION ON POROUS AND FLEXIBLE BREAKWATER 53
3.5 SOLUTION TECHNIQUES 55
3.5.1 Eigenfunction-Expansion Method 55
3.5.2 Wide- Spacing- Approximation Method (WSAM) 56
3.5.3 Least-Squares-Approximation Method 56
4 WAVE SCATTERING BY SURFACE-PIERCING DIKES 59
4.1 INTRODUCTION 59
4.2 MODEL IN THE CASE OF A SINGLE DIKE 60
4.2.1 Definition of the Physical Problem 60
4.2.2 Velocity Potentials 61
4.2.3 General Solution Procedure 65
4.3 MODEL IN THE CASE OF A PAIR OF IDENTICAL DIKES 67
4.3.1 Definition of the Physical Problem 67
4.3.2 Velocity Potentials 67
CONTENTS iii
4.3.3 General Solution Procedure 69
4.4 MODEL USING WSAM 70
4.4.1 Solution Procedure Using WSAM 70
4.5 NUMERICAL RESULTS AND DISCUSSION 72
4.5.1 Reflected Energy 72
4.5.2 Hydrodynamic Forces 74
4.5.3 Summary of Important Observations 76
5 WAVE SCATTERING BY BOTTOM-STANDING DIKES 87
5.1 INTRODUCTION 87
5.2 MODEL IN THE CASE OF A SINGLE DIKE 88
5.2.1 Deflnition of the Physical Problem 88
5.2.2 Velocity Potentials 88
5.2.3 General Solution Procedure 90
5.3 MODEL IN THE CASE OF A PAIR OF IDENTICAL DIKES 91
5.3.1 Deflnition of the Physical Problem 91
5.3.2 Velocity Potentials 91
5.3.3 General Solution Procedure 93
5.4 NUMERICAL RESULTS AND DISCUSSION 94
5.4.1 Reflected Energy 95
5.4.2 Summary of Important Observations 96
6 WAVE PAST POROUS MEMBRANE BREAKWATER 103
6.1 INTRODUCTION 103
6.2 DEFINITION OF THE PHYSICAL PROBLEM 103
6.3 MODEL FOR FLUID FLOW 104
6.4 MODEL FOR MEMBRANE RESPONSE 105
6.5 GENERAL SOLUTION PROCEDURE 106
6.6 NUMERICAL RESULTS AND DISCUSSION 108
6.6.1 Reflected and Transmitted Energy 108
iv CONTENTS
6.6.2 Free Surface and Interface Elevations 110
6.6.3 Response of Membrane Breakwater 112
6.6.4 Summary of Important Observations 113
7 WAVE PAST POROUS PLATE BREAKWATER 125
7.1 INTRODUCTION 125
7.2 DEFINITION OF THE PHYSICAL PROBLEM 125
7.3 MODEL FOR FLUID FLOW 126
7.4 MODEL FOR FLEXIBLE PLATE RESPONSE 127
7.5 GENERAL SOLUTION PROCEDURE 128
7.6 NUMERICAL RESULTS AND DISCUSSION 129
7.6.1 Reflected and Transmitted Energy 130
7.6.2 Response of Plate Breakwater 132
7.6.3 Hydrodynamic Force on Plate Breakwater 132
7.6.4 Summary of Important Observations 133
8 WAVE TRAPPING BY FLEXIBLE POROUS BREAKWATERS 143
8.1 INTRODUCTION 143
8.2 DEFINITION OF THE PHYSICAL PROBLEMS 143
8.3 MODEL FOR FLUID FLOW 144
8.4 MODEL FOR FLEXIBLE PLATE RESPONSE 146
8.5 GENERAL SOLUTION PROCEDURE 148
8.6 NUMERICAL RESULTS AND DISCUSSION 150
8.6.1 The Case of a Bottom-Standing Breakwater 151
8.6.2 The Case of a Surface-Piercing Breakwater 153
8.6.3 Summary of Important Observations 155
9 CONCLUDING REMARKS 173
9.1 SALIENT FEATURES OF THE STUDY 173
9.2 FUTURE SCOPE OF RESEARCH 176
CONTENTS V
BIBLIOGRAPHY 177
LIST OF PUBLICATIONS FROM THE PRESENT THESIS WORK 197
ABOUT THE AUTHOR 199
vi CONTENTS
List of Figures
1.1 Definition sketch for two-layer fiuid wave motion 3
3.1 Definition sketch for gap in case of bottom-standing partial breakwater. . . 49
3.2 Definition sketch for gap in case of surface-piercing partial breakwater. . . 50
4.1 Definition sketch for single surface-piercing dike 61
4.2 Definition sketch for a pair of identical surface-piercing dikes 67
4.3 Comparison of refiection coefficients in SM, Krj and IM, Krjj versus pjd
for a single surface-piercing dike at different H/d values, a/d =1.0, h/H =
0.25 and s = 0.75 with Mei and Black (1969) 77
4.4 Reflection coefficients in (a) SM, Kri and (b) IM, Krji versus pjd for a
single surface-piercing dike at different H/d values, a/d = 1.0, h/H = 0.25
and s = 0.75 78
4.5 Reflection coefficients in (a) SM, Kri and (b) IM, Krji versus pjd for a
single surface-piercing dike at different a/d values, H/d = 6.0, h/H = 0.25
and s = 0.75 79
4.6 Reffection coefficients in (a) SM, Krj and (b) IM, Krjj versus pjd for a
single surface-piercing dike at different h/H values, H/d = 5.0, a/d = 1.0
and s = 0.75 80
4.7 Reffection coefficients in (a) SM, Kri and (b) IM, Kru versus pid for a
single surface-piercing dike at different s values, H/d = 5.0, a/d = 1.0 and
h/H = 0.25 81
vii
viii LIST OF FIGURES
4.8 Reflection coefiicients in (a) SM, Krj and (b) IM, Krjj versus pjd for a
pair of identical surface-piercing dikes at different b/H values, H/d = 6.0,
a/d = 1.0, s = 0.75 and h/H = 0.25 82
4.9 (a) Horizontal force, HF and (b) Vertical force, VF per unit incident wave
amplitude and length of dike in MN/m^ for a single surface-piercing dike
at different H/d values, a/d = 1.0, s = 0.75 and h/H = 0.25 83
4.10 (a) Horizontal force, HF and (b) Vertical force, VF per unit incident wave
amplitude and length of dike in MN/m^ for a single surface-piercing dike
at different a/d values, H/d = 6.0, s = 0.75 and h/H = 0.25 84
4.11 Horizontal force on first, \HFi/Io\ and second, \HF2/Io\ dike in MN/m^
for (a) b/H = 0.25 (b) b/H = 0.75, at H/d = 6.0, a/d = 0.1, s = 0.75 and
h/H = 0.25 85
4.12 Vertical force on first, |V"Fi//o| and second, \VF2/Io\ dike in MN/m^ for
(a) b/H = 0.25 (b) b/H = 0.75, at H/d = 6.0, a/d = 0.1, s = 0.75 and
h/H = 0.25 86
5.1 Definition sketch for single bottom-standing dike 88
5.2 Definition sketch for a pair of identical bottom-standing dikes 92
5.3 Refiection coefficients in (a) SM, Krj and (b) IM, Krn versus pi{H — d) for
a single bottom-standing dike at different H/d values, a/d = 6.0, h/H =
0.25 and s = 0.75 98
5.4 Reffection coefficients in (a) SM, Krj and (b) IM, Kru versus pi{H — d) for
a single bottom-standing dike at different a/d values, H/d = 2.0, h/H =
0.25 and s = 0.75 99
5.5 Reffection coefficients in (a) SM, Krj and (b) IM, Kru versus pi{H — d)
for a single bottom-standing dike at different h/H values, H/d = 2.0,
a/d = 6.0 and s = 0.75 100
5.6 Reffection coefficients in (a) SM, Krj and (b) IM, Kru versus pi(H — d) for
a single bottom-standing dike at different s values, H/d = 2.0, a/d = 6.0
and h/H = 0.25 101
LIST OF FIGURES ix
5.7 Reflection coefiicients in (a) SM, Krj and (b) IM, Krjj versus pjd for a
pair of identical surface-piercing dikes at different b/H values, H/d = 6.0,
a/d = 1.0, s = 0.75 and h/H = 0.25 102
6.1 Definition sketch for fiexible porous membrane breakwater 104
6.2 Reflection and transmission coefficients in (a) SM and (b) IM versus pjH
for different T' values at G = 1 + 2i, s = 0.75 and h/H = 0.5 116
6.3 Reffection and transmission coefficients in (a) SM and (b) IM versus pjH
for different G values at h/H = 0.5, s = 0.75 and T' = 0.4 117
6.4 Reffection and transmission coefficients in (a) SM and (b) IM versus pjH
for different h/H ratios at G = 1 + 2i, s = 0.75 and T' = 0.4 118
6.5 Reffection and transmission coefficients in (a) SM and (b) IM versus pjH
for different s values at h/H = 0.5, G = 1 + 2i and T' = 0.4 119
6.6 (a) Free surface and (b) Interface elevation versus x/Xj for different T'
values at piH = 1.0, h/H = 0.5, G = 1 + 2i and s = 0.75 120
6.7 (a) Free surface and (b) Interface elevation versus x/ Xj for different h/H
ratios at piH = 1.0, G = 1 + 2i, s = 0.75 and T' = 0.4 121
6.8 (a) Free surface and (b) Interface elevation versus x/ Xj for different s values
at piH = 1.0, h/H = 0.5, G = 1 + 2i and T' = 0.4 122
6.9 Membrane displacement versus y/H for different h/H ratios at s = 0.75,
PiH = 1.0, T' = 0.4 and G = 1 + 2i 123
6.10 Membrane displacement versus y/H for different s values at h/H = 0.5,
PiH = 1.0, T' = 0.4 and G = 1 + 2i 123
6.11 Membrane displacement versus y/H for different T' values at h/H = 0.5,
s = 0.75, piH = 1.0 and G = 1 + 2i 124
6.12 Membrane displacement versus y/H for different G values at s = 0.75,
PiH = 1.0, h/H = 0.5 and T' = 0.4 124
7.1 Definition sketch for fiexible porous plate breakwater 126
X LIST OF FIGURES
7.2 Convergence test for reflection and transmission coeflicients in (a) SM and
(b) IM versus piH in case of wave past porous plate breakwater problem
at EI/p2gH^ = 0.01, G = 1, s = 0.75 and h/H = 0.25 135
7.3 Reflection and transmission coeflicients in (a) SM and (b) IM versus pjH
for diflerent EI / p2gH^ values at G = 1, s = 0.9 and h/H = 0.25 136
7.4 Reflection and transmission coeflicients in (a) SM and (b) IM versus pjH
for diflerent G values at h/H = 0.75, s = 0.75 and EI / p2gH^ = 0.02. ... 137
7.5 Reflection and transmission coeflicients in (a) SM and (b) IM versus piH
for diflerent h/H ratios at G = 2, s = 0.9 and EI/p2gH^ = 0.1 138
7.6 Reflection and transmission coeflicients in (a) SM and (b) IM versus pjH
for diflerent s values at h/H = 0.75, G = 1 + 0.5i and EI/p2gH^ = 0.06. . 139
7.7 Breakwater displacement proflle for diflerent EI/ p2gH'^ values at h/H =
0.25, s = 0.9, piH = 0.5 and G = 1 140
7.8 Breakwater displacement proflle for diflerent G values at s = 0.9, pjH =
0.5, h/H = 0.25 and EI/p2gH^ = 0.02 140
7.9 Breakwater displacement proflle for diflerent h/H ratios at s = 0.9, piH =
0.5, EI/p2gH^ = 0.02 and G = 1 141
7.10 Force coefficient versus EI/p2gH^ for diflerent G values at pjH = 0.5,
h/H = 0.25, and s = 0.9 141
7.11 Force coefficient versus EI/p2gH^ for diflerent h/H ratios at piH = 0.5,
s = 0.9 and G = 1 + 0.5i 142
8.1 Deflnition sketch for wave trapping by bottom-standing partial plate break-
water 144
8.2 Deflnition sketch for wave trapping by surface-piercing partial plate break-
water 145
8.3 Reflection coeflicients in (a) SM, Krj versus L/A„ and (b) IM, Krn versus
L/\ii for bottom-standing breakwater at diflerent EI values, h/H = 1.0,
G = 2, h/H = 0.25, and s = 0.75 158
LIST OF FIGURES xi
8.4 Reflection coefiicients in (a) SM, Krj versus L/A„ and (b) IM, Krjj versus
L/ Xii for bottom-standing breakwater at different G values, h/H = 0.25,
b/H = 1.0, EI = 10 Nm, and s = 0.75 159
8.5 Reflection coefficients in (a) SM, Krj versus L/A„ and (b) IM, Krji versus
L/ Xji for bottom-standing breakwater at different b/H values, h/H = 0.5,
G = 2, ^/ = 10 Nm, and s = 0.75 160
8.6 Reflection coefficients in (a) SM, Krj versus L/A„ and (b) IM, Krjj versus
L/Xji for bottom-standing breakwater at different h/H values, b/H = 1.0,
G = 2, ^/ = 10 Nm, and s = 0.75 161
8.7 Reffection coefficients in (a) SM, Kri versus L/A„ and (b) IM, Krn versus
L/\ii for bottom-standing breakwater at different s values, h/H = 0.25,
G = 2, ^/ = 10 Nm, and b/H = 1.0 162
8.8 Barrier deflection, { at (a) h/H = 0.25 and (b) b/H = 1.0 for bottom-
standing breakwater with s = 0.75, G = 1, EI = 20 Nm, and L/Xj = 0.25. 163
8.9 Hydrodynamic force, Fr at (a) s = 0.75 and (b) G = 2 for bottom-standing
breakwater with h/H = 0.25, b/H = 1.0, and L/Xj = 0.25 164
8.10 Reflection coefficients in (a) SM, Krj versus L/Xn and (b) IM, Kru ver-
sus L/Xji for surface-piercing barrier at different EI values, b/H = 0.5,
hi/H = 1.0, G = 2, h/H = 0.5, and s = 0.75 165
8.11 Reflection coefficients in (a) SM, Krj versus L/A„ and (b) IM, Krjj versus
L/ Xjj for surface-piercing barrier at different G values, h/H = 0.5, b/H =
0.7, hi/H = 1.0, EI = 10 Nm, and s = 0.75 166
8.12 Reflection coefficients in (a) SM, Krj versus L/A„ and (b) IM, Krjj versus
L/Xji for surface-piercing barrier at different b/H values, h/H = 0.5, G =
2, hi/H = 1.0, EI = 10 Nm, and s = 0.75 167
8.13 Reflection coefficients in (a) SM, Krj versus L/Xn and (b) IM, Krjj versus
L/Xjj for surface-piercing barrier at different hi/ H values, h/H = 0.5,
G = 2, b/H = 0.5, EI = 10 Nm, and s = 0.75 168
xii LIST OF FIGURES
8.14 Reflection coefiicients in (a) SM, Krj versus L/A„ and (b) IM, Krjj ver-
sus L/Xji for surface-piercing barrier at different h/H values, b/H = 0.5,
hi/H = 1.0, G = 2, EI =10 Nm, and s = 0.75 169
8.15 Reflection coefficients in (a) SM, Krj versus L/A„ and (b) IM, Krjj versus
L/ Xii for surface-piercing barrier at different s values, h/H = 0.25, G = 2,
EI = 10 Nm, hi/H = 1.0 and b/H = 0.5 170
8.16 Barrier deffection, { at (a) hi/H = 1.0 and (b) G = 2 for surface-piercing
breakwater with h/H = 0.5, b/H = 0.7, EI = 10 Nm, s = 0.75 and
L/Xi = 0.25 171
8.17 Hydrodynamic force, Fr at (a) s = 0.75 and (b) hi/H = 1.0 for surface-
piercing breakwater with h/H = 0.25, b/H = 0.7, G = 2 and L/Aj = 0.25. 172
List of Notations
a half of the dike width
6 thickness of the porous medium
Cm added mass coefficient of medium grains
d flexural rigidity
/ resistance force coefficient
Fr hydrodynamic force
G porous-effect parameter
g acceleration due to gravity
H total depth of entire fluid domain
h depth of upper fluid
Iq amplitude of incident wave
//, /// amplitude of incident wave in SM and IM
K = uj^/g
Kri, Krji reflection coefficient in surface mode (SM) and internal mode (IM)
Krj, Krji transmission coefficient in SM and IM
ko incident wave number
Lg vertical gap location
Lif, L^f portion of breakwater in lower and upper fluid
Lop portion of breakwater above free-surface
TUs flexible breakwater mass
m' non-dimensional membrane mass
rii outward normal to the boundary
xni
Pn roots of dispersion relation
pr fluid pressure
Ri, Rii amplitude of reflected wave in SM and IM
s two-layer fluid density ratio
S inertia force coefficient
T tension applied to the membrane
Tj, Tij amplitude of transmitted wave in SM and IM
T' non-dimensional tension parameter
U, V horizontal and vertical fluid velocity
X, y horizontal and vertical Cartesian co-ordinates
P breakwater frequency parameter
7 porosity
(," breakwater deflection (function of y and t)
Vfs, Vint free surface and interface elevation
A/ wave length of incident wave in SM
^ breakwater deflection (function of y)
Pi, p2 density of upper and lower fluid
Ps membrane mass density
$ velocity potential (function of both space and time)
(j) spatial velocity potential
uj wave frequency
XIV
List of Abbreviations
BIE boundary integral equation
DBC dynamic boundary condition
HF horizontal force
IM internal mode
IPBO interface piercing bottom obstacle
IPSO interface piercing surface obstacle
KBC kinematic boundary condition
SM surface mode
VF vertical force
VLFS very large floating structure
WSAM wide-spacing-approximation method
XV
XVI
Chapter 1
INTRODUCTION
Water waves are generated mainly by winds in open seas and large lakes. They carry a sig-
nificant amount of energy from winds into near-shore regions. Thereby they significantly
contribute to the regional hydrodynamics and transport process, producing strong physi-
cal, geological and environmental impact on coastal environment and on human activities
in the coastal area. Furthermore waves are closely connected to the complete dynamics
of coastal systems.
Understanding wave dynamics and predicting wave conditions in harbor and coastal
regions remained an important part of coastal studies over the years. It is particularly
critical to coastal engineering practice in the aspects of, for instance, shoreline protection,
beach erosion, navigation safety, channel maintenance, harbor planning and management,
breakwater and jetty design, etc. Moreover an accurate prediction of the hydrodynamic
effects due to waves interaction with offshore structures is a necessary requirement in the
design, protection and operation of such structures.
1 . 1 TWO-LAYER FLUID
Water waves have been studied for more than three centuries. Since the pioneering work
of Isaac Newton (1687), it is long known that an exact analytical solution of the water
wave problem in general is out of reach. The study and theories dealing with ordinary
2 CHAPTER 1. INTRODUCTION
surface gravity waves are well known and enough information is available on ordinary
surface gravity waves in the open literature. The theories on the water wave propagation
have been developing over the years, and research interest in the recent past appears to
be shifting towards solution of more practical real field problems.
In the recent time, it is observed that there is an increasing interest in understanding
internal-waves because often in an ocean, internal-waves are observed and are the cause of
heavy damages experienced by many onshore and offshore structures. For ocean engineers,
interest in internal- waves is due to their role in submarine detection and the generation of
anomalous drag on ships in fjords and estuaries. Such anomalous drag occurs when fresh
water from rivers and runoff forms a thin layer of light ffuid which lies above the cold
saline water in a narrow fjord. The passage of a ship can then generate internal-waves
which radiate energy away from the neighborhood of the ship. This lost energy is an
additional wave drag for the ship. Earlier the source of this energy loss was not easy to
identify and was regarded as mysterious. Regions having this drag came to be known as
"dead water" regions.
There is one well known theory that the loss of the American nuclear submarine,
"Thresher," in the early 1960s was a result of the effects of a collision with an internal
wave. Internal waves are not only dangerous to submarines, but also can have disastrous
effects on drilling platforms, piers and viaducts. Internal tides cause dramatic vertical dis-
placements of density surfaces in the ocean interior, often several tens of meters and even
hundreds of meters in some locations (Garrett and Kunze (2007)). These displacements,
and the associated currents, complicate the mapping of the average state of the ocean,
and also have major effects on acoustic transmission (Dushaw 2006), sediment transport
(Cacchione et al. (2002) and McPhee-Shaw (2006)), and oil-drilling platforms (Osborne
and Burch 1980, and Osborne et al. (1978)).
Lowering of the salinity of surface waters of the Ocean near a river outffow or thawed
glacier causes deceleration of a ship because internal waves may be generated above the
interface between the low-salinity surface and steeper layers of salty waters. The appear-
ance of "dead water" can have an influence on the maneuvering of submarines. Sailboats
1.1. TWO-LAYER FLUID 3
and towed vessels are often thrown by this phenomenon. The more general scientific inter-
est comes from the general role of internal-waves in energy transport in lakes and oceans.
Moreover, the major challenge in case of internal- waves is that they are not visible to the
naked eye, hence it is difficult to detect them and take precautionary measures.
/ \
(0,0)
X
\__y
Pi
Free Surface
Surface Wave
y=h
\_7
p Interface
"2
Internal Wave
WVVWKWVWVWVWV
y=H
Figure 1.1: Definition sketch for two-layer fiuid wave motion.
The simplest model of an internal- wave is when the density of the liquid is taken to be
piecewise constant. This means, the liquid can be described as two-layers of constant den-
sity over each layer. In many natural bodies of water, stratification of either temperature
or salinity may take place which can lead to significant density differences with depth. A
sharp change in the fluid density at a certain water depth owing to variation in salinity
and/or temperature may be observed in a lake, an estuary or Norwegian fjords. Another
example of sharp density change is a thin layer of muddy water at the bottom of harbors
or channels with relatively shallow water depth. These density changes may signiflcantly
alter the hydrodynamic characteristics of waves past coastal structures. In the present
thesis, attention is restricted to the typical case of an ocean, fjord, or lake where the upper
layer (marked as Fluid 1 in the Fig. 1.1) is a relatively light liquid, and the lower layer
(marked as Fluid 2 in Fig. 1.1) is a relatively heavy liquid. In such stratifled flows a new
type of wave can be observed due to the new mode known as internal- wave.
Two distinct wave modes are possible for the two-layer fluid comprising of a light
liquid, and a heavy liquid. These two modes are known as the surface or fast mode and
the internal or slow mode. The surface or fast mode gives rise to motions which are similar
to those produced by an ordinary surface-wave. Because of the similarity of the mode
shapes, it should not be surprising that the wave speed of the surface or fast mode usually
4 CHAPTER 1. INTRODUCTION
differs very little from the usual gravity wave speed. Internal-waves are generated due
to periodic movements of interface. This motion is usually not observable at the surface
to the naked eyes since it may not effect the surface. The restoring force for internal-
waves is proportional to the product of gravity and the density difference between the
two layers (the relative buoyancy). At internal interfaces this difference is much smaller
than the density difference between air and water (by several orders of magnitude). As
a consequence, internal-waves can attain much larger amplitudes than surface-waves. It
also takes longer for the restoring force to return particles to their average position, and
internal-waves have periods much longer than surface gravity waves. Hence, the speed of
the internal- wave is considerably smaller than the speed of the surface-wave.
In contrast to surface- waves in which horizontal particle velocities are largest at the
surface and either decay quickly with depth (in deep water waves) or are independent of
depth (in shallow water waves), horizontal water movement in internal- waves is largest
near the surface and bottom and minimal at mid-depth. A second distinctive feature of
internal-waves is that the surface disturbance can be extremely small relative to the size
of the disturbances to the internal layer. Even though understanding of such interesting
physical phenomenon in an ocean is very much important in the coastal engineering
applications, very little knowledge on this is available up to date because of the complexity
involved in mathematical formulation and understanding the physics of internal-waves
which propagate simultaneously with the surface-waves in a two-layer fluid.
1.2 WAVE-STRUCTURE INTERACTION
In general wave-structure interaction can be separated into hydraulic responses (such as
wave run-up, wave over-topping, wave transmission and wave reflection), and loads and
response of structural parts. Design conditions for coastal structures includes acceptable
levels of hydraulic responses in terms of wave run-up, over-topping, wave transmission
and wave reflection.
The wave run-up level is one of the most important factors affecting the design of
1.2. WAVE-STRUCTURE INTERACTION 5
coastal structures because it determines the design crest level of the structures in cases
where no (or marginal) over-topping is accepted. Examples includes dikes, revetments and
breakwaters with pedestrian traffic. Wave over-topping occurs when the structure crest
height is smaller than the run-up level. The over-topping discharge is a very important
design parameter because it determines the crest level and the design of the upper part of
the structure. Design levels of over-topping discharges frequently vary, from heavy over-
topping of detached breakwaters and outer breakwaters without access roads, to very
limited over-topping in case where roads, storage areas and moorings are close to the
front of the structure.
With partial breakwaters the incident waves will more or less pass over or under the
structure while retaining much of the incident wave characteristics. In case of imperme-
able flexible structures, wave transmission takes place when the impact of water waves
generates new waves at the rear side of the structure because of the structural deformation.
Permeable structures allow wave transmission as a result of wave penetration.
Design levels of transmitted waves depend on the use of the protected area. Related
to port engineering is the question of acceptable wave disturbance in harbor basins, which
in turn is related to the movements of moored vessels. Coastal structures like breakwaters
reflect some portion of the incident wave energy. If reflection is significant, the interaction
of incident and refiected waves can create an extremely complex sea with very steep waves
that often are breaking. This is a difficult problem for many harbor entrance areas where
steep waves can cause considerable maneuvering problems for smaller vessels. Strong
reflection also increases the seabed erosion potential in front of protective structures.
Hence wave reflection from the boundary structures like breakwaters determines to a
large extent the wave disturbance in harbor basins and maneuvering conditions at harbor
entrances. Moreover, breakwaters can cause reflection of waves onto neighboring beaches
and thereby increase wave impacts on beach processes. In addition, an important part of
the design procedure for structures in general is the determination of the loads and the
related stress, deformations and stability conditions of the structural members.
6 CHAPTER 1. INTRODUCTION
1.3 BREAKWATERS
The destructive power of ocean waves is well known and the methods to provide protection
against these waves have occupied the attention of coastal engineers over the years. Waves
moving over the ocean near the shore may greatly change beaches. Strong winds often
creates a situation, which raises the water level and exposes higher parts of beach to
wave attack which are not ordinarily vulnerable to waves. These waves carry away large
quantity of sand from the beach to the near shore bottom. Land structures, inadequately
protected and located too close to the water, are then subjected to the hydrodynamic
forces of waves and may be damaged and destroyed. Recent rise in damage of both life
and property across the shore lines due to the destructive ocean waves all over the world
further intensified the research to find suitable protection techniques against various type
of wave attacks.
The term wave transmission is used in reference to the wave energy that travels past
a breakwater, either by passing through and/or by over-topping the structure. The wave
energy that is attenuated in the lee of the breakwater is either dissipated by the structure
(due to friction, wave breaking, armor unit movement, etc.) or refiected back as refiected
wave energy. The effectiveness of a breakwater in attenuating wave energy can be mea-
sured by the amount of wave energy that is transmitted past the structure: lesser the
wave transmission coefficient the greater the wave attenuation. Breakwaters are designed
not only to protect the offshore and onshore structures from wave attack but also used to
provide economic protection to harbor, marine and restore eroding beaches.
The type of protection needed is a function of the purpose it has to serve. In gen-
eral, natural or artificial breakwaters of various configurations are used depending on the
applications to provide protection from wave attack in both onshore and offshore areas.
Breakwaters are generally shore-parallel structures and their primary purpose is to reduce
the amount of wave energy reaching the protected area. They are similar to natural bars,
reefs or near-shore islands and are designed to dissipate wave energy. Beaches and dunes
can be protected by an offshore breakwater that reduces the wave energy reaching the
1.3. BREAKWATERS 7
shore. However, offshore structures are usually more costly than onshore structures and
are seldom built solely for shore protection. Offshore breakwaters are constructed mainly
for navigation purpose. A breakwater protecting a harbor area provides shelter for ships
and boats.
Over the years, investigations have been carried out on various types of breakwaters
to find the efficient breakwater suiting a particular application and the coastal region that
needs protection. The major challenge encountered is to select the proper breakwater con-
figuration, material and location so that it will be effective, economical and environment
friendly. For example, in situations where harbors are located in areas where severe wave
conditions occur they often lie sheltered behind one or more breakwaters. The location
of the breakwater should be chosen such that the ships in the harbor will be subjected to
a gentle wave climate.
1.3.1 Rigid Breakwaters
The rigid-fixed breakwaters offer advantage in the form of excellent storm protection.
However, at the same time they contribute several drawbacks to the environment. Careful
thought must be given to the fixed breakwaters: they become an almost permanent part
of the landscape. Hence, if any environmental damage they may cause must either be
accepted or the breakwater must be removed. This may be a very expensive penalty for
a mistake. A rigid-fixed breakwater must not only be carefully designed, but also very
carefully analyzed for its effects on the physical system in which it is to be placed.
Another disadvantage is that a fixed breakwater can be a total barrier to close off a
significant portion of a waterway or entrance channel, thereby causing a faster river or
tidal fiow in its vicinity, as well as potentially trapping debris on the up-drift side. It may
create unacceptable sedimentation and water quality problems due to poor circulation
behind the structure. On the other hand, a detached breakwater may be connected to the
shore by the formation of a tombolo. This could seriously interrupt long-shore transport
and cause down-drift erosion. Although most existing breakwaters are rigid structures
that are fixed to the ocean fioor and projected above the water surface, the shift has
8 CHAPTER 1. INTRODUCTION
been towards more temporary, transportable breakwaters because of the aforementioned
reasons.
1.3.2 Flexible Breakwaters
Recent developments in materials science have contributed significantly to opening an op-
portunity for the fiexible structures as breakwaters. The problem of material degradation
and failure have been largely overcome by modern materials. With the use of modern
materials, fiexible structures might be extremely efiective as breakwater, absorbing or
refiecting much of the wave energy. There has been an increasing interest in the use of
fiexible breakwater as a means of providing protection from wave attack in semi-protected
regions. Furthermore, fiexible breakwaters have generated interest among researchers with
some of its added advantages.
Such structures provide an alternative to more conventional rigid-fixed breakwaters
in areas where poor foundation conditions exist or where protection is required only on
a temporary basis. They can be utilized as temporary sacrificial breakwaters to reduce
the size of storm waves impacting harbors, coastal areas or fixed breakwaters. Another
advantage is that fiexible breakwaters could provide an inexpensive means of protecting
beaches and shorelines exposed to small or moderate waves and offer fast and relatively
easy installation for temporary offshore work. They are reusable, have a lower construction
cost and are lighter in weight compared to the conventional large massive structures. They
can also be more economical compared to fixed-type breakwater especially when the water
depth is large. This is because their cost does not increase substantially with depth, while
the cost of a fixed structure, such as rubble-mound breakwater, increases exponentially
with depth.
Furthermore, fiexible structure can be prefabricated onshore thus reducing construc-
tion time and complexity. They can be moored despite unfavorable seabed soil conditions
and are usually positioned near the water surface where the wave energy is most pro-
nounced. Flexible structures are also attractive because they require only a little mainte-
nance, they can handle extreme temperature and they usually do not corrode. Because of
1.3. BREAKWATERS 9
the aforementioned advantages, flexible breakwaters can support a number of operations
at sea, such as oil and gas extraction, fish farming, ocean mining and recreation. These
structures can also be used for pollution control, salvage operations, and construction and
maintenance of offshore platform. They can serve as a breakwater augmentation device
by reducing the size of waves incident on fixed breakwaters, shores and offshore structures.
In past ffexible plate as a breakwater were more popular (Stoker (1957), Slew and
Hurley (1977), Patarapanich (1978), Cheong and Patarapanich (1992), and Wang and
Shen (1999)). However, in recent studies ffexible wave barrier consisting of a vertically
tensioned membrane have been reported (Kim and Kee (1996), Kee and Kim (1997), Lo
(1998) and Lo (2000)). These refer to structures that mainly consist of membrane made of
synthetic fiber, rubber or a polymeric material. They can be easily fabricated in large size,
allowing for wave control in a wide region. The multi-mode motions of the membrane can
also be explored to widen the effective frequency range for wave attenuation, especially
on irregular waves.
1.3.3 Porous Breakwaters
Impermeable breakwaters have been employed to block waves in harbor for a fairly long
time. Their design is simple and convenient. When they are subjected to wave impact, the
wave load is extremely high because there is almost a total reffection of wave energy. It has
been well-known for a long time that a harbor may be agitated into a resonant state when
subjected to incoming waves of a particular period. Such resonance may lead to extremely
large oscillations of the water surface within the harbor and cause disastrous damage to
its mooring system, especially when the breakwater is a rigid structure. Therefore, care
must be taken in the design to avoid resonance in the harbor. In the case of possible
resonance, wave energy must be radiated or dissipated so that the oscillation can be
effectively suppressed. Recently porous breakwaters are proposed to tackle such difficult
practical problems.
Porous breakwaters can reduce both the transmitted and reffected wave heights. This
kind of structures also experience less hydrodynamic forces and dissipate a significant
10 CHAPTER 1. INTRODUCTION
amount of wave energy. In general flow or wave motion past porous media has strong
relevance in many distinct fields, such as soil mechanics, biology, hydrology and harbor
engineering. It has both academic and practical importance. However, a through un-
derstanding of the problem is still far from being complete, even though its applications
in coastal engineering and harbor engineering is extremely common. Some of the major
applications of porous breakwaters are oil/contaminant spill containment, temporary pro-
tection during coastal construction works and augmentation of existing breakwaters for
seasonal protection. The ability of an engineer to predict wave transmission and refiection
by a porous breakwater plays a central role in the protection of both onshore and offshore
structures. The knowledge about stability of the breakwater under wave attack is also
valuable to harbor engineering.
1.3.4 Partial Breakwaters
Most of the existing breakwaters are in general extended over the entire water depth, i.e
extended from the seabed up to the free surface. However, in recent years, there is a
significant interest in the use of partial breakwaters to attenuate the wave energy. Partial
breakwaters only occupy a segment of the whole water depth. In coastal engineering,
partial barriers as breakwaters are more economical and sometimes more appropriate for
engineering applications. These kinds of breakwaters also provide a less expensive means
to protect beaches exposed to waves of small or moderate amplitudes, and to reduce the
wave amplitude at resonance. These breakwaters can either be suspended or mounted to
the ocean fioor depending on the application and economic constraints. For the first case
they can be fioating or partially submerged and are known as surface-piercing breakwaters.
For the second case they are usually fully- submerged and are known as bottom-standing
breakwaters.
Surface-piercing breakwaters are in general temporary and transportable breakwa-
ters. They are economical alternative to fixed structures for use in deeper waters (at
depth greater than 20 feet) and can effectively attenuate moderate wave heights (less
than about 6.5 feet). Poor soil conditions may make surface-piercing breakwaters the
1.3. BREAKWATERS 11
only options available. They also minimize the interference on water circulation and fish
migration. Further if ice formation presents a problem, surface-piercing breakwaters can
be removed from the site. Surface-Piercing breakwaters are not obstructive and can be
more aesthetically pleasing than fixed structures. They can easily be rearranged in a
different layout and transported to another site for maximum efficiency.
However, in addition to the aforementioned advantages there exist a few disadvantages
of surface-piercing breakwaters. Surface-Piercing breakwaters are ineffective in reducing
wave heights for slow waves. Surface-Piercing breakwaters are susceptible to structural
failure during catastrophic storms. Relative to conventional fixed breakwaters, surface-
piercing breakwaters require a high amount of maintenance. The first surface-piercing
breakwater appeared in 1811 at Plymouth Port in England. During World War II, Bom-
bardon surface-piercing breakwaters were used along the Normandy coast. In 1930, Japan
placed the first surface-piercing breakwater in Aomori Port to test the structure's resis-
tance to waves and the wave dissipation function. At present there are several types
of surface-piercing breakwaters in use, which include box, pontoon, mat, tethered fioat,
fiexible breakwater etc. Most box type breakwaters are reinforced concrete rectangular
shaped modules. These structures have proved to be effective and have a 50 year design
life. However, the main disadvantages for these structures are that they are considerably
more expensive and require higher maintenance than mat and ffexible breakwater type.
In areas where environmental considerations must be evaluated, bottom-standing type
breakwaters are considered more frequently as soft solution in solving coastal engineer-
ing problems. Bottom-Standing breakwaters provide opportunities for environmental en-
hancement, aesthetics (one of the major engineering priorities at the moment) and wave
protection in coastal areas due to their characteristics that are not found in conventional
breakwaters. These characteristics include the ability to promote water circulation and
provide a fish habitant enhancement capacity. The bottom-standing breakwaters are
being used for fish farming in coastal fishery because they create a calm region in the
downstream of the wave motion and act as a sheltered region for a large group of marine
habitats during severe wave conditions. These breakwaters resist the sediment transport
12 CHAPTER 1. INTRODUCTION
and provide a strong protection against coastal erosion. Furthermore, they do not have so
many disadvantages as the hard structures like groins, detached breakwaters, revetments,
seawalls, etc.
Bottom-Standing breakwaters of different types with their crest at or below the still
water level can offer potentially economic solutions in situations where only a partial
protection from waves is required. Bottom-Standing breakwaters are generally less ex-
pensive to construct and maintain and they offer protection to shorelines and beaches in
their natural environment against waves. Further they maintain exchange of water be-
tween the protected area and the open water without hindrance apart from maintaining
aesthetic appearance without impairment. As the bottom-standing breakwater in general
does not extend above the free surface, they cannot be used to stop complete wave motion
in the protected areas but it is sufficient for many practical applications. For bottom-
standing breakwaters the greater the submergence, the lesser the impact of wave energy
on structure, and hence less effective the structure will be for wave attenuation.
1.4 OBJECTIVE OF PRESENT INVESTIGATION
A water wave model that can accurately simulate various aspects of wave transformation
in coastal regions is a valuable engineering tool. Such a tool is very useful in handling
problems like wave scattering/trapping by onshore and off-shore breakwaters. In the
present thesis mathematical models are developed to analyze the water wave scattering
and trapping problems by both rigid and ffexible breakwaters (with and without porosity)
in a two-layer ffuid. Suitable computer codes have been written to solve the various
physical problems. Major attention is given on the following aspects.
• Generalization of single-layer wave structure interactive model to two-layer systems.
• Investigation of surface- and internal wave scattering.
• Evaluation of efficiency of ffexible porous breakwaters in a two-layer ffuid.
• Study of surface and internal wave trapping between the breakwater and a vertical
rigid end wall.
1.4. OBJECTIVE OF PRESENT INVESTIGATION 13
• Evaluating the influence of fluid density and interface location on the effectiveness
of breakwaters in a two-layer fluid.
A class of problems dealing with scattering of harmonic surface- and internal- waves by
various breakwaters, namely (i) Rigid dikes, (ii) Porous membrane and (iii) Porous flexible
plate in a two-layer fluid are studied in the present work. The trapping phenomenon
of surface- and internal-waves by partial porous flexible breakwaters is also investigated.
Under the assumption of the linearized theory, these class of physical problems are reduced
to a class of two-dimensional mixed boundary value problems associated with Laplace's
equation for the determination of the velocity potentials along with the important physical
quantities, like, the reflection and transmission coefficients of an incident time-harmonic
surface-/internal-wave. A general structural response equation is solved in case of flexible
structures like, porous membrane and porous plate, where the response is coupled with
the velocity potential. Eigenfunction-expansion, wide-spacing-approximation (WSAM)
and least-squares-approximation methods are the main mathematical techniques, which
are utilized in the solution of the present mathematical models.
The present Chapter 1 has provided a general introduction, scope and objectives of
the present study. In Chapter 2, an elaborate review of literature and the motivation
for the present investigation are presented. The basic mathematical tools utilized in the
thesis along with the derivations of the basic hydrodynamics and structural response
equations in the linearized set up are elaborated in the Chapter 3. In Chapter 4, surface-
and internal-waves scattering by a single surface-piercing rigid dike is solved numerically
within the context of linearized theory of water waves. After solving this physical problem
the study is extended to a pair of identical rectangular surface-piercing dikes. The surface-
and internal-waves scattering by a single bottom-standing rectangular rigid dike and a
pair of bottom-standing rigid dikes are presented in Chapter 5. Computed results in two-
layer fluid are compared with those existing in the literature for a single-layer fluid. The
results obtained by the matched-eigenfunction-expansion method are compared with that
of WSAM. In Chapter 6 and Chapter 7, surface- and internal-waves scattering by flexible
porous structures is considered. The surface- and internal-wave trapping by porous and
14 CHAPTER 1. INTRODUCTION
flexible partial breakwaters near the end of a semi-infinitely long channel is studied in
Chapter 8. In Chapter 9, the summary and the conclusions of the present investigations
are presented. The scope of the present research work is also discussed. Bibliography is
included separately.
Chapter 2
REVIEW OF LITERATURE
2.1 INTRODUCTION
With developmental activities, the dynamic equilibrium of the coastal region is disturbed,
often resulting in coastal erosion and accretion. Coastal erosion is a severe problem world-
wide threatening the coastal properties, causing degradation of valuable land and natural
resources, disruption to fishing, shipping and tourism. The development of coastal fa-
cilities has necessitated proper management of the sea front warranting construction of
coastal protective structures. The choice of the structure depends on the wave environ-
ment and the morphology of the coastal region. Hence it is very essential to understand
the wave-structure interaction problems to predict the behavior of different breakwaters
at various wave environments.
Wave-structure interaction is a classical problem in coastal engineering. Such a prob-
lem is related to a number of engineering concerns such as structural stability under wave
attack, scour in front of structures, reduction of transmitted wave energy, etc. When
analyzing the problem of waves interaction with breakwaters, the viscous effects are usu-
ally neglected. Therefore, the ffow is analyzed using potential ffow theory. Linear-wave
theory is the core theory and is used extensively in ocean wave modeling over past several
years. It is one of the most useful model which has found wide applications over various
coastal engineering problems. Moreover, the linear-wave theory acts as an stepping stone
15
16 CHAPTER 2. REVIEW OF LITERATURE
for all non-linear theories, which have been developed over the years to solve various diffi-
cult wave mechanics problems. Hence in the coastal engineering applications, linear-wave
theory is often used to solve various physical problems and acts as a foundation for the
non-linear studies. This chapter is divided into two sections. First an extensive review
is carried out describing the earlier works on dynamics of linear surface-waves passing
submerged and floating breakwaters (rigid/flexible/porous breakwaters) in a single-layer
fluid. The second section deals with the review for previous works on the wave structure
interaction in a two-layer fluid within the context of linearized theory of water waves.
2.2 WAVE-STRUCTURE INTERACTION IN A
SINGLE-LAYER FLUID
The dynamics of ocean surface- waves is a topic that has been studied extensively. Some
of the well known results are contained in the books by Dean and Dalrymple (1991), Mei
(1992) and Sorensen (1993). In coastal areas, waves can be partially reflected by rapid
changes in the seafloor. This power of reflection due to variation in bottom depth has led
to numerous studies and the creation of immersed coastal structures designed to protect
the coastal installations or natural shores from wave impact by partial reflection of its
energy (see Rey (1992) and Rey (1995)).
On the other hand, the suspended breakwaters have advantages over those attached to
the seabed, since they can be used for larger depths and therefore are not limited to small
distances from the beach. These type of breakwaters often provide a cost effective and
efficient solution of protection from wave attack in semi-protected regions. They may be
preferred in deeper waters, in areas where poor foundation conditions or environmental
constraints exist or where protection is required only on a temporary basis, such as pollu-
tion control, salvage operations, and construction and maintenance of offshore platforms.
Such structures could also protect ffoating airports and portable ports. In this section the
review for surface-wave interaction with ffoating and submerged breakwaters is carried
out under three subsections namely rigid, ffexible and porous breakwaters.
2.2. WAVE-STRUCTURE INTERACTION IN A SINGLE-LAYER FLUID 17
2.2.1 Rigid Breakwaters
Rigid breakwaters extending to sea-surface have been constructed in the past twenty
years to protect pleasure beaches and shorelines against erosion, especially in Japan.
Recently, the human desire to protect the beauty of shore landscape and the interest of
development of marine recreational areas convinced people to think and apply different
and more environment friendly solutions. One solution, which can meet these needs, is
submerged breakwaters like dikes instead of detached breakwaters. Because of their deep
submergence, ships may pass over them and also the sea-area is available for recreation
purpose. Such structures also could be rapidly deployed to protect a harbor or moored
vessels from the destructive effects of water waves.
Although rigid submerged breakwaters have proven to be less efficient than emerged
breakwaters in reflecting waves, they may be used efficiently as a means of erosion control
(Bruno (1993)). The interaction between water and a submerged marine structure has
attracted increasing attention over the past decade. Not only are the submerged marine
structures related to many ocean and coastal engineering problems, such as dike appli-
cations for preventing beach erosion, but also to the water wave propagation over coral
reefs or continental shelf. Much of the research attention in recent past has focused on
the floating rigid breakwater and there are numerous studies, both experimental and nu-
merical that are reported in the literature. With increasing availability of computational
capability, the complex flow conflguration near the submerged and floating structures can
be analyzed.
Extensive experimental and theoretical investigations have been conducted to exam-
ine the performance of different geometries of bottom-standing rigid breakwaters. Since
the submerged structures are usually used to reduce the transmitted wave, most of the
studies are mainly concerned with determining the reflection and transmission proper-
ties for a given incident wave. For standard geometries, less computationally expensive
methods are available to solve the wave structure interaction problems. An explicit so-
lution for the scattering of waves by a pair of surface piercing vertical barriers in deep
water has been given by Levine and Rodemich (1958). For bottom-standing rectangular
18 CHAPTER 2. REVIEW OF LITERATURE
bodies approximate solutions for fong waves have been developed by Ogilvie (1960), for
long obstacles by Newman (1965), and for low draft structures by Mei (1969). Newman
(1965) obtained an approximate solution for surface- waves elevation in the limit of a long
submerged obstacle.
Levine (1965) studied the interaction of oblique waves with a completely submerged
circular cylinder near the free surface based on the Green's function. Transmission and
reflection coefficients were calculated. When the obstacle is in the form of a thick barrier
with rectangular cross section present in water of uniform finite depth, the corresponding
water wave scattering problems for normal incidence of a wave train have been investigated
by Mei and Black (1969). They use the variational formulation to solve the problem. For
a single fioating cylinder of rectangular cross-section. Black et al. (1971) also used the
variational method to solve the radiation problem (where the body oscillates radiating
waves into otherwise calm water) and then used the Haskind relation (Newman (1976))
to deduce the forces due to incident waves. Free surface elevations were obtained for a
single cylinder and for two cylinders in series. Garrison (1969) investigated the interaction
of an infinite shallow draft cylinder oscillating at the free surface with a train of oblique
waves using the boundary integral method. Wave scattering by a circular dock has been
considered by Garrett (1971).
A number of authors have considered the two-dimensional problems of the radiation
and scattering of waves by two parallel circular cylinders in deep water. Wang and Wahab
(1971) have extended the multipole method of Ursell (1949) to analyze the heaving of two
rigidly connected half-immersed cylinders. Bolton and Ursell (1973) used the multipole
expansion method to the interaction of an infinitely long circular cylinder with oblique
waves. The added mass, damping coefficients and vertical wave force were calculated.
Bai (1975) presented a finite element method to study the diffraction of oblique waves
by an infinite cylinder in water of infinite depth. Refiection and transmission coefficients
and the diffraction forces and moments were computed for oblique waves incident upon a
rectangular cylinder. Raman et al. (1977) reported on the damping action of rectangular
and rigid vertical submerged barriers and expressed the transmission coefficients in terms
2.2. WAVE-STRUCTURE INTERACTION IN A SINGLE-LAYER FLUID 19
of the transmitted energy to the total power of the incident wave. Further it is concluded
that in case of rectangular submerged barrier, the top width of the barrier plays an
important role in controlling transmission coefficient.
An integral equation formulation for the calculation of hydro dynamic coefficients
for long, horizontal cylinders of arbitrary section has been presented by Naftgzer and
Chakrabarti (1979). Due to the general complexity of multi-body problems a number
of authors have used a WSAM where only interactions are assumed to arise from plane
waves traveling between the bodies. The accuracy of this type of approximation has been
demonstrated in a number of cases. Two-dimensional problems considered include two
surface piercing barriers, by Srokosz and Evans (1979) and two half-immersed circular
cylinders by Martin (1984). Martin (1984) has pointed out that the basic assumptions
behind the WSAM are that both the wavelength and a typical body dimension must be
much less than the separation between bodies. It is a remarkable fact that the WSAM
has consistently given good results even when these assumptions are clearly violated.
The test on wave transmission and reflection characteristics of laboratory breakwater
conducted by Seeling (1980) indicate that both the depth of submergence and top width
are important in determining the performance of the breakwater and it is suggested that
for near zero submergence, submerged breakwater is efficient in reducing the transmission.
The absorption of wave energy with a submerged cylindrical duct is studied by Thomas
(1981). Liu and Abbaspour (1982) studied the scattering of oblique waves by an infinite
cylinder of arbitrary shape using a hybrid integral equation formulation and numerical
results were presented. The scattering problem for bodies of arbitrary shape may be
solved by integral equation methods for which an extensive review is given by Mei (1983).
Leonard et al. (1983) extended Bai's (1975) finite element method to solve the diffraction
and radiation boundary value problems arising from multiple two-dimensional horizontal
cylinders interacting with obliquely incident linear- waves. Hydro dynamic parameters and
responses of two cylinders in heave and sway were calculated.
Martin (1984) has solved the scattering problem for two rigidly connected half-immersed
cylinders by the null-field method. Garrison (1984) used a Green's function method to
20 CHAPTER 2. REVIEW OF LITERATURE
compute the oblique wave interaction with a cyhnder of arbitrary section on the free
surface in water of finite depth. The equivalent problem of Levine and Rodemich (1958)
has been solved in finite depth water by Falnes and Mclver (1984) using the method of
matched-eigenfunction-expansions. For barriers of differing length the matching is carried
out at the two boundaries of three adjoining regions. Similar to Naftgzer and Chakrabarti
(1979) an integral equation formulation for the calculation of hydrodynamic coefficients
for long, horizontal cylinders of arbitrary section has been presented by Andersen and
Wuzhou (1985).
Mclver (1986) solved the scattering problem for adjacent ffoating bridges by a di-
rect method. A matched-eigenfunction-expansion method and WSAM is used for the
solution of a number of axisymmetric three-dimensional problems and results from both
the methods are compared. For the case of obstacle in the form of a thick wall with
a submerged narrow gap in finite depth water, Liu and Wu (1987) used the method of
matched asymptotic expansion to obtain an approximate analytical expression for the
transmission coefficient assuming the width of the wall to be of the same magnitude as
the wavelength. Isaacson and Nwogu (1987) developed a generalized numerical procedure
based on Green's theorem to compute the exciting forces and hydrodynamic coefficients
due to the interaction of oblique waves with an infinitely long, semi-immersed fioating
cylinder of arbitrary shape. Numerical results for wave loads and motions were presented.
Williams and Darwiche (1988) analyzed the three-dimensional scattering of waves by el-
liptical breakwaters using eigenfunction-expansions. Their numerical results are valid for
the entire wavelength spectrum and finite obstacle length.
Subcritical and supercritical solutions for surface-waves over circular bumps were given
by Shen et al. (1989). Chakrabarti and Naftzger (1989) evaluated the wave forces on a
submerged semi-cylinder resting on the bottom using a boundary integral method. The
non-dimensional horizontal and vertical forces were obtained at different values of the
wave number. Kobayashi and Wurjanto (1989) presented a numerical approach based
on finite amplitude shallow water equation for determination of wave transmission over
submerged breakwater. They obtained the approximate solution in the limit of long
2.2. WAVE-STRUCTURE INTERACTION IN A SINGLE-LAYER FLUID 21
waves. Design equations for wave transmission over a submerged breakwater are detailed
by Rojanakamthorn et al. (1989) based on mild slope equation and the result suggests that
when the height of the submerged breakwater is 50% of the total depth, the transmission
coefficient varies between 0.4 and 0.7. For the normal incidence of waves most of the
problems can be solved explicitly.
Rey et al. (1992) conducted wave tank experiments over a rectangular submerged
bar. Beji and Battjes (1993) also conducted laboratory experiments to study the gen-
eration of higher harmonics by a submerged trapezoidal bar. MuUarkey et al. (1992)
utilized an eigenfunction-expansion approach to calculate the hydrodynamic coefficients
for rectangular TLP pontoons. Drimer et al. (1992) presented a simplified approach for
a floating breakwater where the breakwater width and incident wavelength are taken to
be much larger than the gap between breakwater and the seabed. Losada et al. (1992)
and (1993) applied the eigenfunction-expansion method to the propagation of oblique
waves past rigid vertical thin barriers and calculated the transmission and reflection co-
efficients. Sannasiraj and Sundaravadivelu (1995) applied the flnite element technique to
study the interaction of oblique waves with freely floating long structures. The hydrody-
namic behavior of two-dimensional horizontal floating structures under the action of the
multi-directional waves has been investigated, and the motions and forces on a rectan-
gular floating structure experiencing unidirectional and multi-directional wave flelds were
computed.
Abul-Azm (1994a) investigated the diffraction through wide submerged breakwaters
under oblique waves by use of the eigenfunction-expansion method and discussed the ef-
fect of different wave and structural parameters on the transmitted and reffected waves
and the hydrodynamic loadings on the breakwater. Mandal and Dolai (1994) used the
one-term Galerkin approximation to determine the upper and lower bounds for the re-
ffection and transmission coefficients in the problems of oblique water wave diffraction by
a thin vertical barrier in water of uniform flnite depth. Mallayachari and Sundar (1996)
investigated numerically the wave reflection characteristics over submerged rectangular
step of flnite length. They compared reflection characteristics of rectangular structures
22 CHAPTER 2. REVIEW OF LITERATURE
with half cyhnder and trapezoidal obstacles. Ertekin and Becker (1996) applied a finite
difi'erence method to examine diffraction of waves by a submerged bottom-mounted trape-
zoidal obstacle. For oblique incidence of the wave trains and/or for finite depth water,
the problems cannot be solved explicitly and they can be tackled by some approximate
methods to obtain numerical estimates for some physical quantities such as the refiection
and transmission coefficients (Sudeshna et al. (1996)).
Isaacson et al. (1996) experimentally investigated the reflection of obliquely incident
waves from a model of rubble mound breakwater. Results show that both the reflection
coefficient and the refiected phase lag are noticeably dependent on the angle of incidence
and that the variation with the angle of incidence further depends on the depth to wave-
length ratio. Abul-Azm and Williams (1997) used the eigenfunction-expansion method to
examine oblique wave diffraction by a detached breakwater system consisting of an infl-
nite row of regular-spaced thin, impermeable structures located in water of uniform depth.
A comprehensive review of wave reflection by uneven bottom has been presented in the
book of Dingemans (1997). The one-term Galerkin approximation was used also by Das
et al. (1997) to evaluate the upper and lower bounds for the reflection and transmission
coefficients in the problem of oblique water wave diffraction by two equal thin, parallel,
flxed vertical barriers with gaps presented in water of uniform flnite depth. Sannasiraj
et al. (2000) used the flnite element method to the study of the diffraction-radiation of
multiple floating structures in directional waves.
Abul-Azm and Gesraha (2000) examined the hydrodynamic properties of a long rigid
floating pontoon interacting with linear-waves in water of flnite depth by use of the
eigenfunction-expansion method. This kind of structure was known as a kind of effec-
tive breakwater and many investigators (e.g., Drimer et al. (1992), Cheong et al. (1996),
Williams et al. (2000) and Zheng et al. (2004)) have studied the radiation and/or diffrac-
tion problem under the action of normal incident waves. Li et al. (2002) explored the
interaction of oblique irregular waves with a vertical wall. The ratio of the oblique wave
forces to the normal incident wave forces was given and the characteristics of the reffection
coefficients for oblique waves were introduced.
2.2. WAVE-STRUCTURE INTERACTION IN A SINGLE-LAYER FLUID 23
Politis et al. (2002) developed a Boundary Integral Equation (BIE) method for oblique
water wave scattering by cylinders in water of infinite depth and four geometrical con-
figurations were chosen to investigate the numerical performance of the BIE method.
The added mass, damping coefficients and excitation forces were calculated. Recently
Soylemez and Goren (2003) studied the diffraction of oblique water waves by thick rect-
angular barrier mounted on seabed. They also studied the diffraction of oblique water
waves by thick rectangular barrier ffoating at the free surface experimentally and investi-
gated theoretically.
The first ever analysis on the problem of trapping of surface-waves over a uniform
sloping beach was carried out by Stokes (1846). This phenomenon of wave trapping
analyzed by Stokes is referred as edge waves which can travel unchanged in the direction
of the shoreline, and decays exponentially to zero in the seaward direction. In spite of
the unbounded fiuid region, this Stokes edge wave is confined or trapped by the sloping
boundary. The understanding of trapped waves in various physical situations is important
in various coastal engineering applications like dynamics and sedimentology of the near
shorezone through their interaction with ocean swells and surfs (Leblond and Mysak
(1978)).
There has been a significant interest in the literature to understand the existence of
trapping waves and many researchers found out the wave frequency corresponding to the
trapped mode in different physical situations within the context of linearized theory of
water waves. Ursell (1951) proved the existence of trapped waves above a submerged
horizontal cylinder of sufficiently small radius in a channel spanning the sidewalls. Jones
(1953) generalized Ursell's result to submerged cylinders of arbitrary but symmetric cross
section in finite water depth. Wave trapping may occur along a vertical wall when there is
an abrupt change in ocean depth (Leblond and Mysak (1978)). The existence of trapped
waves above a submerged horizontal plate was investigated by Linton and Evans (1991)
based on matched-eigenfunction-expansion method. Evans and Kuznetsov (1997) gave
a detailed review of the development in the recent decades that has taken place on the
existence of trapped waves. In all these study, emphasis is given on the wave frequency
24 CHAPTER 2. REVIEW OF LITERATURE
i.e., on the trapped mode.
2.2.2 Flexible Breakwaters
Rigid structures like submerged dikes have major disadvantages like requirement of vast
cross-sectional areas and therefore demanding high construction costs. Furthermore, un-
der most considerations, marine bodies are assumed rigid in the presence of waves and
their elastic deformations are neglected. However, the hydroelastic effect should be con-
sidered under certain wave conditions, as when (i) the body itself is flexible, (ii) the body
is very thin compared to wave parameters, and (iii) the body is very long with respect to
the incident wavelength. The former two cases should be quite obvious. However, in the
latter case, localized deflection or vibration of a long structure becomes significant due
to the continuous excitation of small amplitude waves, although the motion of the whole
body is small as compared to its length.
The interaction of gravitational waves with deformable structures like plate, mem-
brane and ice is of interest in the study of the dynamics of off-shore/on-shore structures,
breakwaters, ice fields and artificial structures (airports and islands) when acted upon
by sea waves. Floating and/or fiexible wave barriers have generated interest amongst
researchers with some of its added advantages. They can be moored despite unfavorable
seabed soil conditions and are usually positioned near the water surface where the wave
energy is most pronounced. They allow for free passage of seawater, fish and sediment
transport beneath, thus being friendlier to the environment. They can also be more eco-
nomical compared to fixed-type breakwater especially when the water depth is large, and
can be prefabricated onshore thus reducing construction time.
A great deal of effort is spent to effectively utilize the ocean space for human activities
and developments. Certain huge platforms, which are in general ffexible are constructed
or extended from shoreline to provide more dry space, while structures like floating ports,
mobile offshore bases are built as working spaces. In recent times, several artiflcial floating
islands are constructed off shoreline. Such huge ffoating structures are categorized as Very
Large Floating Structures (VLFS). Before the construction and positioning of any VLFS,
2.2. WAVE-STRUCTURE INTERACTION IN A SINGLE-LAYER FLUID 25
careful and detailed studies are needed to investigate the hydro dynamic performance and
hydroelastic behavior of not only the VLFS system but also the system of breakwaters
protecting the VLFS.
In the theoretical study for the modeling of wave interactions with flexible struc-
tures, there are two major approaches. These are the mode expansions method and the
eigenfunction-expansions method. In the mode expansions method, the body deforma-
tion is represented by a series of natural modes. The kinematic and dynamic surface
conditions due to elasticity and gravity are treated separately. Based on the mode ex-
pansions method. Bishop and Price (1979) gave a comprehensive summary on the studies
of hydroelasticity of ships, and Gran (1992) summed up the engineering applications of
structural responses of marine structures to waves. Kashiwagi (2000) gave a review of the
developments on VLFS and reported that major works on VLFS are based on the mode
expansions method. However, the mode expansions method is only applicable to a finite
plate. On the other hand, the eigenfunction-expansions method is a more direct method,
as it combines the kinematic and dynamic surface conditions, which give the dispersion
relation satisfied by the wave numbers.
Fox and Squire (1994) used the eigenfunction-expansions method to study the in-
teraction of surface-waves with an ice-covered surface and obtained the solution by the
conjugate gradient method. They observed that the eigenfunctions are not orthogonal
with respect to the conventional inner product, though the eigenfunctions are complete.
Squire et al. (1995) presented an invited review on the interaction of gravity waves with
an ice-covered surface. Balmforth and Craster (1999) developed a method based on the
Fourier transform and Wiener-Hopf technique to study the scattering of gravity waves
incident on an ice-covered ocean and obtained asymptotic and approximate solutions to
the problem. They considered ice as a thin elastic plate in the mathematical model.
To analyze the response of a thin horizontal elastic plate fioating in waves, Ohkusu and
Nanba (1996) combined the kinematic and dynamic surface conditions to obtain the free
surface condition on the plate-covered surface and then solve the problem by the boundary
integral method.
26 CHAPTER 2. REVIEW OF LITERATURE
Sturova (1998) used Fox and Squire's (1994) approach to study the oblique incidence
of surface- waves onto an elastic band. Kim and Ertekin (1998) constructed a complete
set of orthogonal eigenfunctions satisfying the dispersion relations and then obtained the
solution explicitly for predicting the hydroelastic behavior of a shallow-draft VLFS. Nanba
and Ohkusu (1999) analyzed the elastic response in waves of a large floating platform of
thin plate configuration in both shallow water and deep water. The free surface condition
gives an important information regarding the wave numbers. In all the aforementioned
studies related to an elastic plate fioating on the water surface, the plate is assumed to
have a free edge, which suggests that the shear force and the bending moment of the plate
vanish at the edge. However, artificial structures are usually kept fixed or moored at the
edge by ropes, anchors, tension cables, or piles. In such cases, the free edge condition
or the built-in edge condition as per the reality. It may be noted that for the simply
supported edge condition, the defiection and the bending moment are assumed to vanish,
whereas for the built-in edge condition, the defiection and the slope of defiection will
vanish.
In the mode expansion method, Newman (1994) proposed to employ different or-
thogonal polynomials to represent the corresponding mode expansions for different edge
conditions, and claimed that the application of natural modal functions should be lim-
ited to the free edge condition only. He noted that it is very difficult to identify the
fittest modal functions for various edge conditions. Wu et al. (1995) extended Newman's
(1994) idea to analyze the wave-induced responses of an elastic fioating plate. Sahoo et
al. (2001) investigated the interaction of surface- waves with a semi-infinite elastic plate
fioating on the free surface in finite water depth. The hydrodynamic behavior due to
three different types of edge conditions, namely (i) free edge, (ii) simply supported edge,
and (iii) built-in edge is analyzed. They used a newly defined inner product along with
the method of matched-eigenfunction-expansions, to obtain the full solution. The edge
conditions are directly incorporated while using the matching condition along with the
orthogonality property. The inner product defined in their work, is a generalization of the
well-known gravity wave inner product developed by Havelock (1929), which was gener-
2.2. WAVE-STRUCTURE INTERACTION IN A SINGLE-LAYER FLUID 27
alized by Rhodes-Robinson (1979a) and Rhodes-Robinson (1979b) to deal with problems
related to capillary gravity waves.
Submerged flexible breakwaters have received increasing attention recently in the pro-
tection of pleasure beaches and shoreline from erosion. A considerable amount of research
has been devoted to investigate various problems dealing with submerged flexible break-
waters. For coastal zones where water depth increases rapidly the use of concrete caissons
or rubble mound structures as wave-barriers are expensive. In view of this, a horizontal
plate, submerged a flnite depth beneath the sea surface and supported on a group of piles,
that requires less concrete per unit run, is suggested as a possible type of breakwater, in
relatively large water depths (Neelamani and Reddy (1992)). A horizontal single surface
plate flxed at the free surface (Stoker (1957)), a submerged horizontal plate (Siew and
Hurley (1977), Patarapanich (1978), and Wang and Shen (1999)), a group of submerged
plates (Wang and Shen (1999)) and a horizontal double-plate breakwater consisting of a
seaward submerged plate and a leeward surface plate (Cheong and Patarapanich (1992))
are some of the breakwaters used and several theoretical and experimental reports are
available on the hydro dynamic performance of these breakwaters and shore protection
structures.
Number of investigations on different hydrodynamic aspects of horizontal breakwaters
are reported in literature. Since the pioneering study by Heins (1950) on the wave diffrac-
tion from a semi-inflnite submerged horizontal plate in flnite depth and the investigation
by Stoker (1957) on the reflection and transmission coefflcient of long waves propagating
past a surface plate in shallow water, a great deal of effort is spent on obtaining optimum
conditions for minimum wave transmission and obtaining design data of such structures
(Burke (1964), Dick (1968), Hattori and Matsumoto (1977), Siew and Hurley (1977),
Patarapanich (1978), and Wang and Shen (1999)). Evans and Morris (1972) treated the
problem of reffection and transmission of oblique waves by a vertical flxed plate-type bar-
rier in deep water. The method of matched asymptotic expansions has been employed
by Siew and Hurley (1977) and analytical expressions for the reflection and transmission
coefflcients for long waves propagating past a submerged plate in shallow water have been
28 CHAPTER 2. REVIEW OF LITERATURE
obtained.
Patarapanich (1978) has examined the averaged energy flux across various regions
around the submerged plate (with restriction to shallow water case) and has analyzed
the variation of the reflection coefficient with plate length and has derived conditions of
maximum and zero reflection. Wave propagation over a submerged plate has also been
modeled by general numerical methods, such as the boundary integral method developed
by Liu and Iskandarani (1989) and the flnite element method by Patarapanich and Cheong
(1989). Experimental results conducted by Patarapanich and Cheong (1989) show that
the optimum plate width is about 0.5-0.7 times that of the wave length above the plate,
for the plate submergence of around 0.05-0.15 times the water depth, for minimum trans-
mission of waves. Liu and Iskandarani (1991) and Yu et al. (1991a, b) have provided
various information related to possible engineering applications. Neelamani and Reddy
(1992) have experimentally investigated the hydrodynamic performance of a submerged
horizontal plate, for different depths of submergence of the plate and for a wide range
of wave steepness in deep water conditions. Yip and Chwang (1996) have presented an
analytical solution to describe a submerged pitching plate as an active wave controller by
eigenfunction-expansion method.
The above investigations have considered the wave motion over a single submerged
plate. However, there are many occasions, where a submerged plate may not provide
the required wave protection due to the ffow in the relatively longer region between the
plate and the seabed. In view of this, the reffection and transmission coefficients of a
horizontal double-plate system consisting of a seaward surface plate and a submerged
leeward plate have been derived in terms of reffection and transmission coefficients of a
single plate case by Cheong and Patarapanich (1992). Their experimental measurements
show that the optimum degree of submergence of the leeward plate should be about 0.1-
0.2 for minimum wave transmission. The reffection coefficient for the double-plate system
has been observed to be greater than that for a single plate system, for all submerged
depth ratios. Their results suggest a large fraction of reffected wave energy in the double-
plate system. Further, the experimental results show that the transmission coefficient is
2.2. WAVE-STRUCTURE INTERACTION IN A SINGLE-LAYER FLUID 29
a minimum at relative depths of about 0.15-0.20. This range also corresponds to higher
loss coefficients, which tend to remain relatively high even at larger depths. This suggests
that the double-plate combination is more effective than the single plate case at larger
relative depths.
Wang and Shen (1999) have considered wave motion over a group of submerged hor-
izontal plates. The method of eigenfunction-expansions has been applied to obtain the
velocity potentials and the free surface elevations. The unknown constant coefficients have
been determined from the matching conditions, using three sets of orthogonal eigenfunc-
tions. The success of double-plate breakwater developed by Cheong and Patarapanich
(1992) in serving as an effective wave-barrier has motivated Usha and Gayathri (2005) to
consider the case of wave motion over a twin-plate system consisting of a surface plate on
the free surface and a submerged plate of same dimensions just below the surface plate,
for different submergence spacing of the lower plate. The main objective of the study was
to evaluate the hydro dynamic performance of the twin-plate system as a breakwater and
compare its performance with those of other available horizontal breakwaters. The study
was restricted to two-dimensional cases of waves approaching normally to the breakwaters.
The method of eigenfunction-expansions was used to obtain the velocity potentials and
the unknown constant coefficients are determined from the matching conditions, using
three sets of orthogonal eigenfunctions. The eigenfunction-expansion method combined
the kinematic and dynamic surface conditions which give the dispersion relation satisfied
by the wave numbers. This method has been successfully and effectively employed by sev-
eral investigators including Wang and Shen (1999), Sahoo et al. (2001), Fox and Squire
(1994) and Hu et al. (2002) in their analysis of wave motion over submerged structures or
water transmission through a porous medium or on the hydrodynamic performance and
the hydroelastic behavior of the system.
There is a considerable interest among researchers over past several years to analyze
the effectiveness of vertical barriers in attenuating wave energy. A huge system of ffap-
type barriers has been planned in Italy to protect the lagoon of Venice against high tides
(Natale and Savi (1993)). Bai (1975) studied the reflection and transmission coefficients
30 CHAPTER 2. REVIEW OF LITERATURE
and the diffraction forces and moments for oblique waves incident upon a vertical flat
plate. Studies using linear theory for their reflection power as a function of wavelength
have shown the presence of maxima and minima in the reflection (Patarapanich (1984)
and Sturova (1991)).
The solution of the flexible emerged breakwater problem has been treated dynamically
as a one-degree-of-freedom system (see, Leach et al. (1985) for a single hinged flap and
Sollitt et al. (1986) for double flaps). Evans and Linton (1990) suggested that submerged
floating breakwaters may be tuned to the incoming waves to provide more reflection than
emerged, surface-piercing structures. They had treated their system as a single-degree-
of-freedom system. The effect of the breakwater rigidity had been considered by Lee and
Chen (1990) for a hinged flap breakwater. Effects of ffexibility and buoyancy had been
introduced by Williams et al. (1991) for one compliant beam-like clamped breakwater
using an appropriate Green function, and by Abul-Azam (1994) for a double flxed or
hinged flap using an eigenfunction approach.
Williams et al. (1992) had extended the numerical solution of Williams et al. (1991)
to a submerged compliant and clamped breakwater problem. They used an integral
equation method, which includes a laborious technique for flnding an appropriate Green's
function. This method is very much useful and computationally attractive when more
complicated geometries are considered (Natale and Sivi (1993)). Williams (1993) analyzed
the wave diffraction due to a pair of ffexible breakwaters consisting of compliant, beam-like
structures, also anchored to the sea bottom and kept under tension by a small buoyancy
chamber at the top. Abul-Azm (1995) presented an analytical solution for the submerged
breakwaters, based on expressing the linear hydrodynamic wave potential in terms of
eigenseries expansion. This expansion technique had been shown to be a computationally
efficient method for several linear and non-linear problems (see, Abul-Azm (1993)). The
breakwater is considered to be ffexible beam-like, either clamped or hinged at the seabed,
anchored to the seabed by tethers and kept under tension by means of a ffoating buoy at
the breakwater tip. The assumptions are the same as used by Williams et al. (1992).
Recently, ffexible barriers consisting of vertically tensioned membranes spanning the
2.2. WAVE-STRUCTURE INTERACTION IN A SINGLE-LAYER FLUID 31
entire water depth were reported. The flexible membrane, which is light, inexpensive,
reusable, and rapidly deployable, can be used as a portable and sacrificial breakwater.
Since it can be easily removed, it is considered as having minimum environmental impacts
on various coastal processes. They can be easily fabricated in large sizes, allowing for wave
control in a wide region. The multi-mode motions of the membrane can also be explored
to widen the effective frequency range for wave attenuation, especially on irregular waves.
Kim and Kee (1996) and Kee and Kim (1997) reported results for a single membrane with
the latter incorporating the effects of a buoy. Both the eigenfunction and the boundary
integral method were used in their studies.
Using the eigenfunction approach, Lo (1998) and Abul-Azm (1994b) investigated the
cases of the dual ffexible membrane and dual hinged beams respectively, while Cho et
al. (1998) used the boundary integral method for dual membranes tensioned by buoys.
Lo (2000) further considered a ffexible membrane of finite length with gaps between the
membrane and the water surface and/or the bottom. In the studies for a membrane
system without buoys, a typically fixed boundary condition is applied at the upper edge
of the vertical membrane that coincides with the water surface. In practice, the tension
on the membrane will have to be applied through buoys or through cables above the
water surface. The effect of tides also implies that a membrane will have to protrude
above the mean water surface in order to intercept the waves at high tide conditions. The
scattering effect of buoys have been examined numerically and experimentally by Kee and
Kim (1997) and Cho et al. (1998) in their studies using the boundary integral method.
Cho et al. (1997) studied the performance of ffexible membrane wave barriers in oblique
incident waves. The performance of surface-piercing or submerged buoy /membrane wave
barriers is tested with various membrane, buoy, and mooring characteristics and wave
conditions including oblique wave headings.
Chan and Lee (2001) investigated wave characteristics past a ffexible fishnet. Lee
and Lo (2002) reported the effect of having the membranes protrude above the water
surface such as that rising from having tension provided by frame with surface cables.
They considered an array of vertical membranes of arbitrary draft and protrusion. The
32 CHAPTER 2. REVIEW OF LITERATURE
eigenfunction-expansion method is used assuming linear- wave theory and small membrane
motion. A mixed dynamic boundary condition results at the plane of the membrane and is
solved by the method of minimal squares. Computational as well as experimental results
are presented for single and dual membrane arrangements. Viscous energy loss is also
incorporated for better comparison of the experimental data with numerical prediction.
In the recent past there have been attempts to solve and understand the complex
physical problems such as analyzing the wave trapping by means of flexible structures.
Using the method of hypersingular integral equations Parsons and Martin (1995) inves-
tigated the trapping of water waves by submerged plates in water of infinite depth and
discussed the results of inclined fiat plates and for curved plates that are symmetric with
respect to a line drawn vertically through their centers.
2.2.3 Porous Breakwaters
Recently porous breakwaters are reported to be a preferred choice as wave attenuat-
ing breakwaters for various coastal engineering applications in the literature. There are
widespread applications of porous structures in coastal engineering. A porous structure
allows the partial transmission of water waves with energy dissipation. Meanwhile, the
wave heights in front of the porous structure and the wave-induced forces on the structure
can also be reduced. However, the porosity of the breakwater presents a challenge to any
attempt at a predictive model. This is because porosity can change the complexity of the
fiow in a number of ways during various wave conditions.
The theoretical study of wave motion through a porous medium depends on estab-
lishing a general theory of porous medium fiow. However, even in its simplest form, this
physical phenomenon is extremely complicated. A common theme of research on the
surface-waves and porous medium interaction during the past decade has been on the
understanding of (a) wave motion past a porous medium, (b) a porous medium as an
active wave controller and (c) stability of a porous medium under wave attack. In fact,
this subject has found applications not only in the harbor but, more importantly, in the
laboratory, where testing and experiments of designs are carried out before they are put
2.2. WAVE-STRUCTURE INTERACTION IN A SINGLE-LAYER FLUID 33
into actual use.
In past there were many attempts to understand the water wave interaction with
various porous structure configurations. It is found that using variable permeability allows
the determination of results for solid obstacles with any number of holes. One classic
example of such problem is that considered by Ursell (1947), who investigated waves
past a surface-piercing barrier reaching partway to the bottom in infinitely deep water.
Considerable effort has been devoted to achieving a good understanding of the phenomena
of harbor resonance. Jarlan (1961) was the first to study the use of a perforated vertical
wall breakwater to reduce wave energy. Significantly progress in the study of harbor
resonance was made by Miles and Munk (1961) and Le Mehaute (1961). Lewin (1963)
studied the refiection of waves by a barrier with any number of gaps in infinitely deep
water. His solutions were complicated and no numerical results or concrete examples were
presented. The studies conducted by Dick and Brebner (1968) on solid and permeable
type submerged breakwaters indicate that infinitely long porous structures with near zero
submergence are capable of reducing the incident wave energy by 50%. In finite depth,
the problem of Ursell (1947) has been studied numerically by Mei and Black (1969).
Rigorous solutions to harbor resonance problem were also presented by Lee (1971), who
considered rectangular and circular harbors with opening located on a straight coastline.
Although many important aspects of the resonance phenomenon can be understood by
considering a geometrically idealized harbor, any practical application depends on the
establishment of numerical theories or codes that are able to deal with arbitrary harbor
configurations (Chwang and Chan (1998)). Such theories are available after Lee (1971).
Tuck (1971) and Porter (1972) derived formal solutions for the transmission of water
waves through a thin plate with a small gap in infinite water depth using potential theory.
Guiney et al. (1972) extended Tuck's (1971) theory to incorporate a finite barrier thickness
and obtained similar results, both theoretically and experimentally. They found that
at very low frequency the thickness of the barrier has little effect. However, at and
above the frequency of maximum energy transmission, the point where the transmission
is maximum, the effect of thickness on reducing energy transmission is pronounced. The
34 CHAPTER 2. REVIEW OF LITERATURE
frequency at which maximum transmission occurs also tends to be slightly lower for thick
barriers.
Le Mehaute (1972) also studied a wave motion past vertical plates placed parallel
to the direction of flow to act as a wave absorber in a wave flume. These plates cause
partial transmission and reflection of the incident wave because of the sudden variation
of permeability as well as because of the increase of laminar boundary layer. The idea
is actually similar to a thick barrier with multiple slits. In most of existing studies it is
assumed that the porous medium obeys Darcy's law. For wave motion across a porous
medium, Sollit and Cross (1972) gave a modification to Darcy's law that was later used
extensively by many workers in the field. In their approach, the force exerted by the porous
medium on the fiow was assumed to be composed of two components; (a) a resistance
force, linearized through an interaction procedure (described below) in accordance with
Lorentz's principle of equal work, and (b) an inertial force, which is linearly proportional to
the fiuid acceleration and is represented by an added mass coefficient. The concept of Sollit
and Cross (1972) is also used in the famous Morison equation by Sarpkaya and Isaacson
(1981) to predict wave forces on small rigid bodies. Raichlen (1974) reviewed some of
the experiments on the effect of waves on rubble-mound structures and provided much
experimental evidence and information about the characteristics of porous structures as
breakwaters.
Madsen (1974) worked out the wave transmission and reflection characteristics of a
long wave train past a single porous medium, assuming the value of inertial coefficient as
one. The theoretical work of Madsen (1974) also show reasonable agreement with those of
other theories and experimental works like those of Kondo and Toma (1972), Nasser and
McCorquodale (1975), and Qui and Wang (1996). Analyzing the same problem as that
of Le Mehaute (1972) using a slightly different approach. Tuck (1975) obtained equiv-
alent results. Mei and Chen (1975) developed the hybrid-boundary-element technique
for harbor engineering. Macaskill (1979) made a comprehensive analysis and numerical
study of the wave motion past a thin permeable barrier. His method involves the use of
Green's function and can be generalized to study the flow past a barrier with any number
2.2. WAVE-STRUCTURE INTERACTION IN A SINGLE-LAYER FLUID 35
of slits at different locations, which is convenient for computational analysis. Macaskill's
(1979) and Tuck's (1975) wave characteristics results in the presence of a porous plate
show some similarity, although not exactly equivalent trend. Macaskill (1979) derived an
empirical formula to relate the geometric permeability to that of Darcy's permeability.
However, essential differences between the definition of permeability or porosity exist be-
tween Macaskill's (1979) analysis and Darcy's law, and this makes reconciliation difficult.
Although this first approximation can be used in an attempt to compare the two results,
the formulae are only valid for a certain range of permeability and cannot be regarded as
in general agreement (Chwang and Chan (1998)). When a porous medium is itself subject
to oscillation, it can also behave as an active generator for wave motion. The study of
this subject was initiated by Madsen (1970). He analyzed the infiuence of leakage around
a piston-type wavemaker and found that the leakage effect was great in reducing the wave
amplitude. The porous effect on the wavemaker can find applications in the study of
surface-waves in reservoirs or lakes caused by landslides during earthquakes.
Another possible application of a porous wavemaking mechanism is in situations where
the efficiency of the generation of waves is of interest. A porous wavemaker may be
helpful in reducing the total load that is accompanied by a reduction of wave amplitude.
Chwang (1983) produced a comprehensive analysis of a thin porous wavemaker. Chwang's
(1983) theory was extended by Chwang and Li (1983) to analyze the waves generated by
a porous wavemaker near the end of a semi-infinitely long channel of constant depth.
Chakrabarti (1989) used an integral transform method to extend Chwang's (1983) theory
when the effect of surface tension is also taken into consideration. His results show that
surface tension reduces the wave amplitude slightly but does not greatly affect the wave
characteristics. Chwang and Dong (1985) studied analytically the wave motion past a
thin vertical porous plate in an infinite or semi-infinite wave fiume. They assumed that
the porous plate is thin hence following Taylor (1956), who proposed that the seepage
velocity can be assumed to be linearly proportional to the pressure difference across the
two sides of the plate.
Evans (1990) studied theoretically the use of multiple porous screen as a means to
36 CHAPTER 2. REVIEW OF LITERATURE
damp the waves in a narrow wave tank. He employed Tuck's (1975) empirical modification
to include both inertial and viscous effects. In their proposed empirical equations, the
proportionality constant between the pressure jump and normal velocity is analogous to
the acoustic impedance in studies of acoustics phenomena as mentioned by Morse and
Ingard (1968). Evans's (1990) results for a single screen are in general identical to those
obtained by Chwang and Dong (1985) for a single plate. He found that the distance
for optimal wave absorption also depends on the wave number and porous coefficient.
Dalrymple et al. (1991) studied the reffection and transmission of a wave train at an
oblique angle of incidence by an infinitely long porous structure with a finite thickness.
Huang (1991) extended Chwang's (1983) theory to accommodate finite thickness as
well as the inertial of a porous plate. Based on Blot's theory of poroelasticity, Huang and
Chao (1992) studied the inertial effect of the porous structure. They located a surface of
seepage at the interface between the porous structure and the ffuid region; that is, the
wave profiles at the two regions are discontinuous. This discontinuity causes the velocity
to change abruptly at the interface. This velocity singularity has also been noted by many
previous workers (Muskat (1946), Richey and Sollitt (1970) and Madsen (1983)) but is
usually neglected because of the difficulty of analysis and because of its limited region of
influence. Hybrid-boundary-element technique was used by Su (1993) to study harbors
of arbitrary geometry. Besides direct applications to the harbor engineering field, the
theories of fiow past porous medium also have important connections to many surface-
wave laboratory experiments. Wang and Ren (1993) presented a theoretical study on the
scattering of small amplitude waves by a fiexible, porous and thin beam like breakwater
held fixed in the seabed. Huang et al. (1993) corrected the earlier solution and also
performed a boundary integral approach to verify the validity of a thin porous wavemaker.
Yu and Chwang (1994a) investigated the resonance in a harbor with porous breakwa-
ters with the wave entering at an arbitrary angle. They derived a new boundary condition
for the porous breakwaters, which includes the complex porous effect parameter. This
modified parameter include both inertial and resistance effects on the porous medium. If
the resistance effect in the porous medium is more prominent, the complex porous effect
2.2. WAVE-STRUCTURE INTERACTION IN A SINGLE-LAYER FLUID 37
parameter reduces to the Chwang parameter (Chwang (1983)). They presented a detailed
study on the phenomena of wave induced oscillation in a harbor with porous breakwa-
ters. They concluded that the inertial effect of the porous structure is mainly to increase
the resonant wave number and it does not reduce the amplitude of the resonant oscilla-
tion effectively, while the porous resistance effect reduces the amplitude of the resonant
oscillation effectively and gives a little change of the resonant wave number.
Yu and Chwang (1994b) also investigated the wave behavior within the porous medium.
They found that as long as the medium impedance is purely imaginary, the porous medium
is nondissipative. This confirms the fact that there is no decay of progressive waves in
a nondissipative medium. They also note that increasing the absolute value of the reac-
tance of a nondissipative medium has the same effect as increasing the wave frequency
or water depth. For slightly dissipative medium, the reactance dominates the resistance.
Yu and Chwang (1994b) found that the first-order modification to the progressive wave
is a stationary decaying wave with the same feature as evanescent waves in water with
no porous medium. This wave causes the leading-order progressive wave to decay. The
first-order modifications to the evanescent waves, on the other hand, are progressive waves
that emit the leading-order stationary evanescent waves. An interesting phenomenon oc-
curs for a strongly dissipative medium. Yu and Chwang (1994b) found that increasing
the medium resistance causes shallow-water waves to be shortened and deep-water waves
to be elongated. Besides the direct impact of waves on a porous medium, a considerable
amount of work has also been devoted to the diffraction of water waves by a submerged
horizontal porous plate. The motive behind this is the possibility of using the submerged
porous plate as an alternative to the breakwater, since a submerged breakwater does not
visually partition the sea and allows the free exchange of water or marine animals for the
benefit of the environment. Factors effecting the reffection, transmission, energy loss and
force on the plate include the dimensionless length of the plate, the relative water depth,
the submergence depth of the plate and the porosity of the plate.
Yu and Chwang (1994c) employed the boundary integral method to study wave diffrac-
tion by a horizontal porous plate submerged at a certain distance below the free surface
38 CHAPTER 2. REVIEW OF LITERATURE
in a fluid of constant depth. Chwang and Wu (1994) extended the study to consider wave
diffraction by a porous disk, although the essential features of the wave characteristics
remain similar. It is observed that the reflection coefficient varies periodically as the
length increases. This is a common feature of wave propagation over the submerged solid
obstacles (Mei and Black (1969)) and is largely due to the change of celerity when waves
propagate in different water depth. Recently, there has been a great deal of effort directed
towards quantifying wave interaction with porous ocean structures.
Wave diffraction by a semi-porous cylindrical breakwater protecting an imperme-
able circular cylinder was investigated theoretically by Darwiche et al. (1994) by an
eigenfunction-expansion approach. The interaction of linear water wave in a channel of
constant depth, impinging on a vertical thin porous breakwater with a semi-submerged
and flxed rectangular obstacle in front, is investigated by Yang et al. (1997). Williams
and Li (1998) extended this analysis to deal with the case where the interior cylinder
is mounted on a large cylindrical storage tank. Using a least-squares method, Lee and
Chwang (2000a) studied the scattering and generation of water waves by vertical per-
meable barriers. Eigenfunction-expansion method was used by Twu et al. (2002) to
examine the wave damping characteristics of vertically stratifled porous structures un-
der oblique wave action. It was found that the wave damping efficiency of a vertically
stratified porous structure behaves very similar to a simple structure for common angle of
incidences and the refiection coefficient decreases with increasing angle of incidence while
the transmission coefficient only slightly increases as the angle of incidence increases.
Till date there are only a few studies on wave trapping by porous structure. Sahoo
et al. (2000) studied the trapping and generation of surface-waves by submerged vertical
permeable barriers or plates kept at one end of a semi-infinitely long channel of finite depth
for different barrier and plate configurations. Recently Yip et al. (2002) investigated the
trapping of surface-waves by submerged vertical porous and fiexible barrier near the end
of a semi-infinitely long channel of finite depth. In these problems, emphasis is given on
the wave characteristics in the trapped region i.e., between the channel end- wall and the
barrier.
2.3. WAVE-STRUCTURE INTERACTION IN A TWO-LAYER FLUID 39
2.3 WAVE-STRUCTURE INTERACTION IN A
TWO-LAYER FLUID
In all the aforementioned linear water wave studies, free surface- waves in a fluid of constant
density over the entire fluid domain are considered. However, waves can also exist at
the interface between two immiscible liquids of different densities. Such a sharp density
gradient can, for example be generated in the ocean by solar heating of the upper layer,
or in an estuary or a fjord into which fresh (less saline) river water flows over oceanic
water, which is more saline and consequently heavier. The situation can be idealized as
two-layer fluid by considering a lighter fluid of density pi lying over a heavier fluid of
density p2- In the case of a two-layer fluid having an interface and a free surface, two
different propagating modes may be excited during the wave motion. The waves generated
due to the presence of the free surface are referred to as surface modes (SM) whilst the
waves generated due to the presence of the interface are referred to as internal modes
(IM) (see Milne-Thomson (1996) and Kundu and Cohen (2002)). The waves in both the
modes propagate simultaneously in the two-layer fluid. This makes the mathematical
formulation and analysis quite complex. This may be the probable reason for negligible
progress in two-layer fluid studies. A review on linear water wave studies in a two-layer
fluid is presented in this section subsequently.
The propagation of waves in a two-layer fluid with both a free surface and an interface
(in the absence of any obstacles) was flrst investigated by Stokes (1847) and the classical
problem of this type of two-layer fluid separated by a common interface with the upper
fluid having a free surface is given in Lamb (1932) Art. 231 and Wehausen and Laitone
(1960). Until recently, very little work has been done on wave/structure interaction in
two-layer fluid. Sturova (1994) approximated the free surface as a rigid lid and studied the
radiation of waves by an oscillating cylinder, moving uniformly in a direction perpendicu-
lar to its axis. Linton and Mclver (1995) developed a general theory for two-dimensional
wave scattering by horizontal cylinders in an inflnitely deep two-layer fluid. They derived
the reciprocity relations that exists between the various hydro dynamic characteristics of
40 CHAPTER 2. REVIEW OF LITERATURE
the cylinders.
It is well-known that a circular cylinder submerged in an infinitely-deep uniform fiuid
reflects no wave energy, and it was shown in Linton and Mclver (1995) that this is also
true for a cylinder in the lower layer of a two-layer fluid. Zilman and Miloh (1995, 1996)
analyzed the effect of a shallow layer of fluid mud on the hydrodynamics of floating bodies.
Sturova (1999) considered the radiation and scattering problem for a cylinder both in a
two-layer as well as in a three-layer fluid bounded above and below by rigid horizontal
walls. For the three-layer case the middle layer was linearly stratifled representing a
smooth pycnocline. Using the method of multi modes Sturova was able to calculate the
hydrodynamic characteristics of the cylinder. Gavrilov et al. (1999) investigated the
effects of a smooth pycnocline on wave scattering for a horizontal circular cylinder where
the fluid is bounded above and below by rigid walls. Their work included a comparison
between theoretical and experimental results, with reasonable qualitative agreement but
notable quantitative disagreement. A similar approach is to assume that the pycnocline is
very thin and model the interface between the two fluids as a sharp discontinuity between
layers of constant density. In the absence of obstacles, the appropriate dispersion relation
for such a two-layer fluid has two solutions for a given frequency (Lamb (1932), Art. 231).
One of these solution corresponds to waves where the majority of the disturbance is close
to the free surface and the other to waves on the interface between the two fluid layers.
Work on three-dimensional scattering can be found in Yeung and Nguyen (1999) and
Cadby and Linton (2000). In the former study, an integral equation technique was em-
ployed to solve radiation and diffraction problems for a rectangular barge in flnite depth,
whereas in the latter study, multi-pole expansions were used to solve problems involving
submerged spheres in water of inflnite depth. The symmetry relations for the added-mass
and damping matrices and an analogue to the Haskind relations were given in Yeung and
Nguyen (1999). Lee and Chwang (2000b) studied the wave transformation by a vertical
barrier between a single-layer fluid and a two-layer fluid. By using the linear-wave theory
and eigenfunction-expansions the boundary value problems are solved by a suitable ap-
plication of the least-squares method. The deflnitions of the corresponding reflection and
2.3. WAVE-STRUCTURE INTERACTION IN A TWO-LAYER FLUID 41
transmission coefficients are introduced in each of the cases that they have considered. It
is found that water waves, propagating either from the homogeneous or from the two-layer
fluid, are partially reflected or transmitted and produced simultaneously in both modes
(SM and IM) of water waves in the two-layer fluid.
In Barthelemy et al. (2000), the scattering of surface- waves by a step bottom in a two-
layer fluid was considered. This problem is of particular interest to understand how tides
are scattered at the continental shelf break. A WKBJ technique, which approximates
the solution by simple traveling waves locally, was employed to find the refiection and
transmission coefficients of the surface-waves past the step. A complete derivation of
reciprocity relations for three-dimensional scattering in two-layer fiuids can be found in
Cadby and Linton (2000). The motivation for their work came from a plan to build
an underwater pipe bridge across one of the Norwegian fjords. In the Norwegian fjords
typically, bodies of waters consists of a layer of fresh water of about 10 m thick lies on
the top of a very deep body of salt water. An extension model by Linton and Cadby
(2002) to oblique waves of the work of Linton and Mclver (1995) in which a linear water
wave theory concerning the interaction of oblique waves with horizontal circular cylinder
in either the upper or lower layer are also solved using multipole expansions. However,
very little progress has been made on wave interaction with fiexible/porous structures in
two-layer fiuid.
Ahmed (1998) considered the case of two immiscible layers of incompressible fiuids
in the presence of a porous wave maker immersed vertically in the two fiuid, which are
periodic in the horizontal direction direction along the wave maker as well as in time. An
asymptotic analysis for large values of time and distance is given for the depression of
the free surface and the surface of separation between the two fiuids. His results indicate
that there are two modes of waves spreading at each of the free surface and surface of
separation. Also his results justify the use of Sommerfeld radiation condition at infinity
when investigating steady state harmonic surface in wave problems. The application
of this condition instead of boundedness condition at infinity is necessary to render the
solution unique. Sherief et al. (2003) investigated the two-dimensional steady linear
42 CHAPTER 2. REVIEW OF LITERATURE
gravity wave motion in two immiscible layers of incompressible and nonviscous fluid in the
presence of a porous wave maker immersed vertically in the two fluids, where the surface
of the upper fluid is free. Manam and Sahoo (2005) tackled analytically the problem of
waves past rigid porous structures in two-layer fluid by making use of the generalized
orthogonal relation. They obtained complete analytical solutions for the boundary value
problems corresponding to the generation or scattering of axi-symmetric waves by two
impermeable and permeable co-axial cylinders.
There are only a few studies on internal- wave trapping. Greenspan (1970) found
solutions for coastal trapped surface- and internal-waves in a continuously and uniformly
stratified fiuid on a sloping beach.
2.4 MOTIVATION FOR THE PRESENT INVESTI-
GATION
The objective of the present investigation is to develop a simple, economic and accurate
mathematical models for predicting the wave and structural behaviors for the wave struc-
ture interaction problems in a two-layer fiuid. The investigation also aims at studying
the performance of various submerged and fioating breakwaters (rigid, fiexible, porous,
partial breakwaters) in attenuating the surface- and internal-waves in a two-layer fiuid.
Until very recent years, the research works on wave-structure interaction problems
in a two-layer fiuid have been found very rarely. Again, the propagation of surface-
and internal-waves in a two-layer fiuid presents a challenge not only in developing a
mathematical model but also in analyzing the complex fiow physics. This is the reason
why there is a negligible progress in the two-layer fiuid studies. All these have prompted
the present investigation.
Chapter 3
GENERAL MATHEMATICAL
FORMULATION
3.1 INTRODUCTION
Knowledge of waves and the forces they generate is essential for the design of coastal
projects since they are the major factors that determines the geometry of beaches, plan-
ning and design of marinas, waterways, shore protection measures, hydraulic structures,
and other civil and military coastal works. Estimation of wave conditions are needed in
almost all coastal engineering studies. To study the water wave propagation in the ocean
over the years many theories were developed. The most elementary wave theory is the
small-amplitude or linear wave theory. This theory, developed by Airy (1845), is easy to
apply, and gives a reasonable approximation of wave characteristics for a wide range of
wave parameters. Although there are limitations to its applicability, linear theory can
still be useful provided the assumptions made in developing this simple theory are not
grossly violated. The assumptions made in developing the linear wave theory applied in
this work are:
1. The fluid is homogeneous and incompressible; therefore, the density is a constant.
2. Surface tension can be neglected.
3. Coriolis effect due to the earth's rotation can be neglected.
43
44 CHAPTER 3. GENERAL MATHEMATICAL FORMULATION
4. Pressure at the free surface is uniform and constant.
5. The fluid is ideal or inviscid (lacks viscosity) and the flow is irrotational.
6. The wave amplitude is small.
7. The seabed is horizontal, fixed and impermeable.
8. Waves are long-crested, i.e., two-dimensional.
9. The particular wave being considered does not interact with any other form of water
motion like current.
The first three assumptions are valid for virtually all practical situations. It is necessary
to relax the fourth, fifth and ninth assumptions for some specialized problems. Relaxing
the sixth, seventh and eighth assumptions is also acceptable for many problems related
to coastal engineering. In the present thesis linear wave theory is used to solve various
wave-structure interaction problems in a two-layer fluid. Hence present mathematical
model is based on all the above assumptions.
Mathematical modeling for solving wave-structure interaction problems in a two-layer
fluid involves not only a particular way of writing the equations of fluid flow and structural
motion but also selection of suitable numerical techniques to solve those equations. The
purpose of this chapter is to introduce the mathematical model for fluid flow and structural
response along with the various solution techniques that are used in the present thesis. In
mathematical formulation and solution techniques the emphasis is given on those aspects
that are speciflc to the problems dealing with fluid structure interaction in a two-layer
fluid considered in the work.
3.2 MATHEMATICAL MODEL FOR TWO-LAYER
FLUID
The simplest model of an internal-wave involves two layers of constant density fluids.
Here the attention is restricted to the typical case of an ocean, fjord, or lake where the
upper layer of density pi has a relatively light liquid and the lower layer of density p2 is a
3.2. MATHEMATICAL MODEL FOR TWO-LAYER FLUID 45
relatively heavy liquid (see Fig. 1.1 in Chapter 1). The general physical and mathematical
model of the two-layer system is similar to that used in the analysis of water waves in a
single-layer fluid. Because the densities are taken to be piecewise constant, the condition
of incompressible flow is made in each liquid. Furthermore, the condition of irrotational,
inviscid flow is also reasonable for each layer so that the governing equations in Fluid 1
and 2 can be taken to be Laplace's equation and the incompressible, irrotational, unsteady
version of Bernoulli's equation.
The physical basis of the boundary conditions at the bottom and interface between
Fluid 1 and 2 is also essentially the same as in the classical case. At a rigid bottom,
only the kinematic boundary condition (KBC) is required. At the interfaces between the
various fluids, both the kinematic and dynamic boundary conditions are required. Surface
tension is normally negligible so that the dynamic boundary condition (DBC) reduces to
the condition of zero pressure jump across the interfaces. An issue which is normally
not faced in the study of ordinary water waves is that KBC imposes constraints on the
velocities on both sides of the interface between Fluids 1 and 2. In a two-layer fluid, KBC
must be applied explicitly on each side of the interface.
The analysis of the resultant system of equations is essentially the same as the classical
water wave problem. In two-layer fluid dispersion relation the frequency can be found
as a function of wavenumber (or equivalently, the wavelength), the thicknesses of each
liquid layer, the ratio of the densities of fluids 1 and 2, and the acceleration of gravity.
These roots of the dispersion relation represent right and left moving surface-like waves
and right and left moving internal- waves. In the present section governing equations and
boundary conditions for wave motion in a two-layer fluid are described subsequently.
3.2.1 Definition of Velocity Potential and Governing Equation
Consider a case where 2D wave propagate in the x direction, and y is the vertical co-
ordinate, which is taken on -ve direction here for convenience (see Fig. 1.1). The free
surface displacement is r}fs{x,t). The fluid domain is assumed to be of uniform finite
depth. According to the assumptions made earlier, a velocity potential ^{x,y,t) exists
46 CHAPTER 3. GENERAL MATHEMATICAL FORMULATION
such that ^{x,y,t) = Re[(()o(()(x,y)exp(—iujt)], where oj is the wave frequency. The factor
(po = —iglo/oj is removed for convenience in the construction of velocity potentials con-
taining the eigenfunctions, where /q is the amplitude of the incident waves and g is the
gravitational constant. The Governing Equation for the problem is specified by Laplace
equation for 0:
V20 = ^+^ = O (3 1)
dx"^ dy"^
3.2.2 Linearized Free Surface Boundary Conditions
At the free surface, the KBC specify that the fiuid particle never leaves the surface, that
is
jj^ - 4>y, at y = r]f^, (3.2)
where D/Dt = d/dt + U{d/dx) is the material or substantial derivative, and 0^, is the
vertical component of fiuid velocity at the free surface. Eq. 3.2 then can be written as:
(% + (/^) =#) . (3.3)
For small-amplitude waves, both U and drjfg/dx are small, so that the quadratic term
U{dr}fg/dx) is one order smaller than other terms in Eq. 3.3, which then simplifies to
^ at ''y=Vf« ^oy''y=Vfs
This condition can be simplified further by noting that the right hand side can be evaluated
at y = rather that at the free surface. To justify this, d(j)/dy is expanded in a Taylor
Series around y = 0.
Therefore, to the first order of accuracy desired here, d(j)/dy in Eq. 3.4 can be evaluated
at y = 0. We then have
^^|, a. ..0. ,3.)
In addition to the KBC at the free surface, there is a DEC that the pressure just below
the free surface is always equal to the ambient pressure, with surface tension neglected.
3.2. MATHEMATICAL MODEL FOR TWO-LAYER FLUID 47
Taking the ambient pressure to be zero, the surface boundary condition is
pr = 0, at y = r?/^. (3.7)
Since the motion is irrotational, the following form of Bernoulli's equation is applicable:
^ + ^r + ^^) + ^-gy = i^W, (3.8)
where g is the gravitational constant. The negative sign appears in the body force term
as the y coordinate is positive vertical in downward direction (see Fig. 1.1). The function
F{t) can be absorbed in d^/dt by redefining $ and Eq. 3.8 can be simplified by neglect-
ing nonlinear term ([/^ + V^) for small-amplitude waves, and the linearized form of the
unsteady Bernoulli equation is
9$ pr
— + --gy = 0. 3.9
ot p
Substituting into the surface boundary condition Eq. 3.7 in Eq. 3.9 gives
-Q^ - gVfs = 0, at y = rjf^. (3.10)
As explained before, for small-amplitude waves, the term d^/dt can be evaluated at y =
rather than at y = rjfg to give the following linear DBC:
— -gVfs = 0, aty = 0. (3.11)
The KBC and DBC given by Eqs. 3.6 and 3.11 can be combined to give a single linear
free surface condition:
-^ + K(/) = 0, ony = 0, (3.12)
oy
where K = oj'^/g.
3.2.3 Linearized Interface Boundary Conditions
Following the similar approach as in the case of free surface boundary condition we can
obtain the conditions at interface for small-amplitude waves. At the interface {y = h),
48 CHAPTER 3. GENERAL MATHEMATICAL FORMULATION
the continuity of the vertical component of velocity and pressure yield the boundary
conditions (see Wehausen and Laitone (I960))
{^+ K(^)y = H^ = S(^ + K(^)y = h_, (3.14)
where s is the density ratio in a two-layer fluid and is defined as s = p\/ P2 with < s < 1.
3.2.4 Boundary Condition on the Rigid Boundaries
The condition on any rigid boundaries is given by
I^^O. (3.15)
where n^'s are the outward normals to the boundaries.
Boundary Condition on the Seabed
The boundary condition on the seabed (y = H) which is assumed horizontal, fixed and
impermeable is given by:
-^ = 0, at y = H. (3.16)
dy
3.2.5 Radiation Conditions at Infinity
In addition to all the above boundary conditions the velocity potential must satisfy the
radiation conditions corresponding to refiected and transmitted waves. The radiation
conditions for the wave motion in a two-layer fiuid are given by
//
(/)^^(/„ei^"^ + i?„e-i^"^)/„(p„,y) as x ^ -00, (3.17)
n=I
and
//
<p^Yl Tne'^-'^fniPu, y) as X ^ +00, (3.18)
n=I
where //, i?/, Tj and ///, Rn, Tjj are the incident, refiected and transmitted wave am-
plitudes in surface mode (SM) and internal mode (IM) respectively. It may be noted that
3.2. MATHEMATICAL MODEL FOR TWO-LAYER FLUID
49
Pj and pjj are wave numbers for the incident waves in SM and IM respectively. fn{Pn, y)'s
are the unknown eigenfunctions which have to be determined by solving the governing
equation (Eq. 3.1) and imposing the appropriate boundary conditions according to the
physical problem under consideration. Similar definitions for the velocity potentials in a
scattering problem for a two-layer fiuid are given by Barthelemy et al. (2000).
3.2.6 Continuity Conditions Across the Gap
flexible Dreakwater
Free surface
y =
Interface
y = H-b
y = h
y = H
Figure 3.1: Definition sketch for gap in case of bottom-standing partial breakwater.
Across any imaginary boundary separating the two fiuid regions (see, Figs. 3.1 and
3.2) over/underneath the breakwater (reforred to as as gap, and the location is defined
by y G Lg) the continuity of velocity and pressure gives
901 902
)i = 02 and -—- = -—- on y G Lg.
ox ox
(3.19)
Here 0i and 02 represent the upstream and downstream velocity potentials respectively
across the gap.
50
CHAPTER 3. GENERAL MATHEMATICAL FORMULATION
nd ^
reakwater
y = -(h,-H)
y =
Free surface
Interface
y = h
y = H-b
y = H
Figure 3.2: Definition sketch for gap in case of surface-piercing partial breakwater.
3.3 MATHEMATICAL MODEL FOR BREAKWA-
TER RESPONSE
The vertical fiexible breakwater response is analyzed by assuming that breakwater be-
haves like a one-dimensional beam of uniform fiexural rigidity EI, axial force T and mass
per unit length m.,. It is assumed that breakwater is defiected horizontally with displace-
ment Ciy^t) = ^6[{(y)e~"''^*], where {(y) represents the complex defiection amplitude and
is assumed to be small as compared to the water depth. In the present thesis work both
the cases of the plate and membrane breakwater are solved. In this section the gov-
erning equations and boundary conditions are presented for both plate and membrane
breakwaters.
3.3.1 Governing Equation
For a Plate
The governing equation of the vertical plate breakwater response is given by
3.3. MATHEMATICAL MODEL FOR BREAKWATER RESPONSE
51
0, on y G Lop,
i^Pi(0i - 02),
on y G L„j,
iwP2(01 - 02),
on y G Lif.
(3.20)
For a Membrane
The governing equation of the vertical membrane breakwater response is given by
(iy2
T-^ + m,uj'i = {
0, on y G Lop,
ia;pi((/)i - 02), on y e L^f, (3.21)
i^P2(0i - 02), on y G L;j,
where L„j and L;j represent the parts of the breakwater in upper and lower fluid domain
respectively. L^p represents the part of the breakwater above the free surface. 0i and
02 represent the upstream and downstream velocity potentials respectively across the
breakwater.
3.3.2 Edge Conditions
For a Plate
At the clamped edge of the breakwater, the vanishing of the displacement and slope of
deflection yield
{Oy=CE = 0, and {(')y=cE = 0. (3.22)
At the free edge, the vanishing of bending moment and shear force give rise to
(C").=F£ = 0, and {e")y=FE = 0.
(3.23)
For a Membrane
At the clamped edge of the breakwater, the vanishing of the displacement yield
iOy=CE = 0.
(3.24)
52 CHAPTER 3. GENERAL MATHEMATICAL FORMULATION
At the free edge, the vanishing of slope of deflection give rise to
{ny=FE = 0, (3.25)
where CE and FE represents the location of clamped and free edges of different break-
water configurations.
3.3.3 Continuity Condition Across the Free Surface
For different regions the breakwater response is defined as below (see, Figs. 3.1 and 3.2).
ay)
^opiy), for x = 0, ye Lop,
^ufiy), for X = 0, ye L^f, (3.26)
6/(y), for x = 0, y G Lif.
For a Plate
Across the free surface (y = 0), the continuity of plate breakwater defiection, slope of
plate breakwater defiection along with the bending moment and shear force acting on the
plate breakwater (Yip et al. (2002)) yield
Uo) = ^uf{o), C(o) = C/(o), C(o) = C'/(o), C(o) = C(o)- (3-27)
For a Membrane
Across the free surface (y = 0), the continuity of membrane breakwater defiection along
with the slope of membrane breakwater defiection yield
eop(O) = C„/(0), C(0)=C/(0). (3.28)
3.3.4 Continuity Condition Across the Interface
For a Plate
Across the interface {y = h), the continuity of plate breakwater defiection, slope of plate
breakwater defiection along with the bending moment and shear force acting on the plate
3.4. CONDITION ON POROUS AND FLEXIBLE BREAKWATER 53
breakwater yield
^^f{h) = ^if{h), cAh) = eif{h), e:f{h) = e/f{h), cw = c,7w. (3.29)
For a Membrane
Across the interface (y = h), the continuity of membrane breakwater deflection along with
the slope of membrane breakwater deflection yield
(ufih)=(ifih), Cfih) = Cifih). (3.30)
3.4 CONDITION ON POROUS AND FLEXIBLE
BREAKWATER
In the present study the boundary condition on the vertical porous breakwaters is derived
following the steps of Yu and Chwang (1994a), which is a generalization of the one devel-
oped by Chwang (1983). Here, the porous-effect parameter (Chwang (1983)) also called
the Chwang parameter (Lee and Chwang (2000a)) is a complex number, which includes
both the inertia and resistance effects.
Within the porous breakwater, the fluid flow is assumed to be followed by governing
equations presented by Sollitt and Cross (1972) (see also Dalrymple et al. (1991)). As
the breakwater is assumed to be flexible hence the equation of motion in the horizontal
directions has the following form:
where U is the time- dependent horizontal velocity vector of the fluid, C{y,t) is the time-
dependent response of the flexible porous breakwater, VPr is the hydrodynamic pres-
sure gradient, p is the fluid density, / is the linearized resistance, oj is the angular fre-
quency and 5 = 1 + ((1 — j)/j)Cm is the coefficient of the inertial force acting on the
porous medium with 7 denoting the porosity and Cm the added mass coefficient. In the
54 CHAPTER 3. GENERAL MATHEMATICAL FORMULATION
present analysis it is assumed that (,", U and Pr are sinusoidal with respect to time, i.e.,
{(, U, Pr) = {^,u,pr)e~^''^^, and by applying this in Eq. 3.31 we obtain:
Vpr 1
u =
ia;^, (3.32)
P ^if-^s)
which indicates that the horizontal velocity is proportional to the pressure gradient.
The porous breakwater is assumed as a thin structure in the present study and hence,
the variation of pressure across its thickness may be approximated as linear. The hori-
zontal velocity component u within the porous medium can thus be expressed in terms of
the pressure jump from one side of the porous wall to another, that is
luji, (3.33)
pMJ - \s)
where pvi and pr2 represents the upstream and downstream pressure across the porous
breakwater respectively and h is the physical thickness of the porous structure.
Since the linearized Bernoulli equation gives
pr
P
(3.34)
and the continuity law across the breakwater leads to a relation
we finally have
7^1,2 = -7^ = -^—, (3.35)
ox ox
''' = iA;oG((/)i - 02) + ic^C, (3.36)
dx
where ko is the incident wave number and
^ = kobiP + S^) =^'^ ^^" ^^-^^^
is a complex porous-effect parameter as defined by Yu and Chwang (1994a). If the re-
sistance is predominant in the porous medium, that is, S <^ f , G = Gr = j/{kobf) is
purely real. Eq. 3.36 then coincides with the formula used by Chwang (1983), which
was based on the assumption that the fiow in the porous medium is governed by Darcy's
law and the porous structure is rigid. On the other hand, if the inertial effect in the
3.5. SOLUTION TECHNIQUES 55
porous medium is more important, that is, f <^ S, we have G = iGi = ij/{kobS), which
is a purely imaginary value. Under this circumstances, Eq. 3.36 is consistent with that
derived by Macaskill (1979) on the basis that the Bernoulli equation holds for the porous
medium flow. Generally, both the resistance and the inertial effect are important. The
parameters / and S may be empirically evaluated following Madsen (1974). However,
for accuracy hydraulic experiments are necessary to estimate the values of / and S for
different breakwater configurations. Some results on the same can be found in Li (2006).
In the present study above derived complex porous-effect parameter G is used in the
boundary condition (Eq. 3.36) across the porous breakwater.
3.5 SOLUTION TECHNIQUES
To solve any ffuid structure interaction problem, selection of proper solution techniques
and computational tools suiting the problem under consideration is one of the major task.
In the present work, the following three solution techniques are utilized for solving the
aforementioned boundary value problems.
3.5.1 Eigenfunction-Expansion Method
The governing equation for all problems considered here is the Laplace equation. Solu-
tions of Laplace equation are called harmonic functions. Apart from water wave modeling
Laplace equation finds its application in many areas like heat conduction problems, me-
chanics, electromagnetism, probability, quantum mechanics, gravity and biology. The
eigenfunction-expansion method is a powerful computational tool for solving boundary
value problems governed by Laplace equation. Eigenfunction-expansion method is par-
ticularly being utilized extensively in the water wave modeling for past several decades.
Generally in a water wave modeling problems this method is used to construct the veloc-
ity potentials, which also provides the dispersion relation. In general dispersion relations
have more then one solutions and these solutions are known as eigenvalues. Depending
upon the physics of problem, necessary eigenvalues are included in the expression of the
56 CHAPTER 3. GENERAL MATHEMATICAL FORMULATION
velocity potential. The functions in the velocity potential that contain these eigenvalues
are known as eigenfunctions. Eigenfunction-expansion method is utilized to solve different
physical problems considered in the subsequent chapters of the present work.
3.5.2 Wide-Spacing-Approximation Method (WSAM)
In general wave scattering by multiple bodies is a complex problem. To simplify the prob-
lem in the past many researchers have utilized the wide-spacing-approximation method
(WSAM) to solve multi-body problems. In this method it is assumed that the only inter-
actions occur arise from plane waves traveling between the bodies. Accuracy of this type
of approximation has been demonstrated by a number of authors for several cases (Srokosz
and Evans (1979), and Martin (1984)). The basic assumptions in a WSAM is that both
the wavelength and a typical body dimension must be much less than the separation dis-
tance between the bodies (Martin (1984)). It is quite interesting to note that WSAM has
consistently provided good results even though these assumptions are clearly violated in
many situations. In the present study WSAM is utilized to solve the multi-body problems
and the results are compared with those obtained with eigenfunction-expansion method.
3.5.3 Least-Squares-Approximation Method
In general for potential flow problems in water wave modeling equations with series ex-
pansion in orthogonal polynomials are encountered. For such equations the evaluation of
coefficients by least-squares methods is significantly simpler. The values of the expansion
coefficients are independent of the point of truncation of the series. For example consider
a series expression obtained in the form
oo
^ Rnfn{y) = 0, for a domain, (3.38)
n=l
where Rn and fn{y) are complex coefficients and functions with complex variables respec-
tively. We can then write
N
Q{y) = J2Rnfn{y). (3.39)
ra=l
3.5. SOLUTION TECHNIQUES 57
Applying the least-squares method and integrating over the defined domain we can obtain
rUL
/ |Q(y)p(iy = minimum. (3.40)
Jll
where UL and LL represents the upper and lower limit for the integral. Minimizing the
above integral with respect to each Rn's yields
rUL
LL
Qiy)Q*dy = 0, (3.41)
dQ{y) - ^ _ _
where Q*{y) = — , Q{y) = X/ ^nfniy) with bar denoting the complex conjugate and
oRm „=i
Rm = Ri, i?2, •••, Rn- Least-squares method for the complex coefficients is expressed
as above is also known as conjugate-gradient method. The expression (3.41) provides
A^ linear equations with N unknowns, which can be solved easily by Gauss-elimination
method. N can be selected by satisfying a convergence criteria based on the desired order
of accuracy.
In the case of fiexible breakwaters the number of unknowns are more than that of rigid
structures. Numerical convergence problem is observed, when eigenfunction expansion
method is employed directly for fiexible breakwaters. Hence, in the present study the
least-squares method is employed to solve the problems dealing with fiexible breakwaters.
58 CHAPTER 3. GENERAL MATHEMATICAL FORMULATION
Chapter 4
WAVE SCATTERING BY
SURFACE-PIERCING DIKES
4.1 INTRODUCTION
In the present chapter wave interaction with rigid non-porous structures of standard
geometry in a two-layer fluid is considered. The case of surface-piercing dikes are analyzed
and the results are compared with those, that exist in the literature for the same problem
in a single-layer fluid. Wave scattering by such floating structures are considered in past
by many researches within the context of linearized-theory of water waves. Some of the
important studies in the context includes Mei and Black (1969) and Mclver (1986). The
mathematical formulation, solution techniques and numerical results for the scattering of
surface- and internal- waves by a single and a pair of surface-piercing dikes in a two-layer
fluid are presented here.
Cartesian co-ordinate system is chosen with the positive x— axis in the direction of
wave propagation and positive y— axis in the downward direction. The dikes are approxi-
mated as cylinders of rectangular cross section and are placed in a two-layer fluid of flnite
depth. These rectangular dikes are partially immersed in a two-layer fluid. The problem
is considered in two-dimensions under the assumptions that the dikes have parallel gen-
erators and are of sufficient length for normal waves incidence. The upper fluid has a free
59
60 CHAPTER 4. WAVE SCATTERING BY SURFACE-PIERCING DIKES
surface (undisturbed free surface located at y = 0) and the two fluids are separated by a
common interface (undisturbed interface located at y = h), each fluid is of infinite hori-
zontal extent. The upper fiuid of density pi occupies the region — oo < x < oo; < y < h
and the lower fiuid of density p2 occupy the region — oo<a;<cx);/i<y<i7in. The
fiow is simple harmonic in time with angular frequency oj. Thus the velocity potential
^{x,y,t) can be expressed as: ^{x,y,t) = Re[((){x,y)exp{—iujt)].
The spatial velocity potential will satisfy the Laplace equation (Eq. 3.1) along with
the appropriate boundary conditions (linearized free surface boundary condition Eq. 3.12,
linearized interface boundary conditions Eqs. 3.13 and 3.14, boundary conditions on the
rigid boundaries Eq. 3.15 and seabed condition Eq. 3.16), which depends on the region of
the fiuid under consideration. In addition radiation conditions at infinity must be imposed
for uniqueness of the solution.
4.2 MODEL IN THE CASE OF A SINGLE DIKE
4.2.1 Definition of the Physical Problem
In this subsection, solution of wave scattering by a single surface-piercing dike is analyzed
which is further generalized to study the wave scattering by a pair of identical surface-
piercing dikes in the subsequent section. Wave scattering by a single surface-piercing dike
as shown in Fig. 4.1 is considered, which is of width 2a and mean wetted draft d. The
origin (0, 0) is chosen to be the point where the mean free surface intersects with the
vertical axis passing through the center of the dike.
The symmetry of the configuration can be exploited to simplify the solution by writing
the velocity potential (f){x,y) as a sum of symmetric and anti-symmetric parts (similar to
Mclver (1986)) as given by
(i> = (t>' + r, (4.1)
where the symmetric velocity potential, 0* and anti-symmetric velocity potential, 0° are
even and odd functions of x respectively. With this decomposition of velocity potential.
4.2. MODEL IN THE CASE OF A SINGLE DIKE
61
x=-a x=0 x=a
y=0
/////////
////////
X
y=d
(i)
^
y=h
(ii)
y=H
y?
P2
wwwwwwwwwwwwwwww'wwwwwwwwwww
Figure 4.1: Definition sketch for single surface-piercing dike.
the boundary value problem reduces to a simpler problem in the region a: > only, and
in addition the number of unknowns to be determined by solving a linear matrix system
is also dramatically reduced.
4.2.2 Velocity Potentials
Considering the waves incident from large positive x upon the dike, the velocity potentials
are obtained by eigenfunction-expansion method in each of the two regions (i) and (ii) as
marked in Fig. 4.1. This is a boundary value problem specified by equation (3.1) along
with the conditions, Eqs. (3.12), (3.13, 3.14, 3.15 and 3.16). Let us take (f){x,y) as
(l){x,y)=X{x)Y{y).
Substituting (j){x,y) in Eq. (3.1) the problem reduces to
(fX
= p X; and
d^Y
= p^Y.
(4.2)
(4.3)
dx'^ ' dy'^
Considering the three different possibilities p < 0, p = and p > and suitably applying
the boundary conditions we obtain the velocity potentials for different regions (regions (i)
and (ii)). The velocity potentials are presented in symmetric and anti-symmetric parts
in the subsequent subsections.
The expression for symmetric velocity potential 0* in an open water region is given by
1
//
>i = ^ Z! (^wi-iPuix - a))f^{y) + ^ A^exp(ip„(a; - a))f^{y), in region (i). (4.4)
^ n=I n=IJIA
62 CHAPTER 4. WAVE SCATTERING BY SURFACE-PIERCING DIKES
The corresponding expression for the anti- symmetric vefocity potential 0" is given by
ill CO
(j)i = -Yl exp(-ip„(a; - a))f^{y) + ^ A°exp(ip„(a; - a))f^{y), in region (i). (4.5)
"^ n=I n=I,II,l
In the open water region (i), the vertical eigenfunctions /„'s, satisfying governing
equation (3.1) along with the conditions Eqs. (3.12, 3.13, 3.14) and Eq. (3.16) on the
seabed y = H , are given by
-A^-^sinh p^{H - h) [p^cosh p^y - Ksinh p^y]
p^sinh Pnh — Kcosh p^h
fn(y) = { in = I, II, 1, 2, 3, ...)
A^~"^cosh Pn{H — y), for h < y < H,
(4.6)
where
^n = {4:Pn)~^{K cosh Pnh - Pnsmh p„/i)"^[(Kcosh p^h - p^sinh p^hf{2{H - h)pn+
sinh 2pn{II — h)) — s sinh^ Pn{H — h){K^{2pnh — sinh 2p„/i)
-pl{2p^h + sinh 2p^h) + 2Kp^{cosh 2p^h - 1))]. (4.7)
In the above pj and pu are positive real roots and Pn for n > 1 are positive purely
imaginary roots of the dispersion relation in p given by
(1 — .s)p^tanh p{H — h) tanh ph — pK[tanh ph + tanh p{H — h)]
+K^[s tanh p{H - h) tanh ph + I] = 0. (4.8)
Here pi and pn represent the propagative modes for the surface- and internal- waves
respectively.
The expression for symmetric velocity potential 0* in the dike covered region is given
by
COS Q X
02 = 5oXo(y) + E K —Xn{y), in region {it). (4.9)
^t7^^ cos g„a
The corresponding expression for the anti-symmetric velocity potential 0" is given by
BnXYoiy) ^ sin q„x , , , , , ,
02 = + E K-^^Xniy), in region (ii). 4.10
a J , sm fl„a
n=l,l ^"'
4.2. MODEL IN THE CASE OF A SINGLE DIKE 63
For this dike covered region (ii), the eigenfunctions satisfying Eqs. (3.1, 3.13, 3.14
and 3.16) and Eq. (3.15) at the bottom of the dike {y = d), are given by
Lo\ hi d<y <h,
Xo{y) = { (4.11)
s Lq ^, for h < y < H,
and
-L;;^sinh qn{H - h) cosh g„(y - d)
Xn{y) =
for d < y < h,
where
sinh Qnih — d)
in = I, 1,2,3, ...) (4-12)
L;;^cosh qn{H - y), for h <y < H,
Ll = s{s{H -h) + {h-d)), (4.13)
and
Ll = (4g„)-^(sinh q^ih - d))-^[s sinh^ q^iH - h){2{h - d)qn + sinh 2g„(/i - d))
+sinh^ qjh - d){2{H - h)q^ + sinh 2q^{H - h))]. (4.14)
In the above qi is positive real and q^s for n > 1 are positive purely imaginary roots
of the dispersion relation in q as given by
(1 — s)q tanh q{H — h) tanh q{h — d) — Kftanh q{h — d)
+s tanh q{H - h)] = 0. (4.15)
It may be noted that in the dike covered region, in case oid < h, the vertical eigenfunc-
tions have only one propagating mode because of the presence of the interface, whereas in
case of d > h (the special case of surface obstacle), the vertical eigenfunctions x^'s do not
satisfy the interface conditions Eqs. (3.13 and 3.14). Like in a single-layer fluid, in this
later situation no propagative mode exists. Hence, similar to the case of a single-layer
fluid problem the term n = I does not appear in the expressions of eigenfunctions and
the corresponding vertical eigenfunctions becomes
Xn = L-' cos q^{H-y), (n = 0, 1, 2, ...), (4.16)
64
CHAPTER 4. WAVE SCATTERING BY SURFACE-PIERCING DIKES
where
Ll = l- d/H, and L^ = 0.5(1 - d/H), for (n > 1)
Qn satisfy the relation
Qn = n7T/{H — d), for n = 0, 1, 2,
(4.17)
(4.1^
The radiation conditions which physically states that scattered wave must be out going
from the dike require that (Barthelemy et al. (2000))
II
(p r^ ^[exp(-ip„a;)/„(y) + {A^ + A^) exp(-2ip„a) exp(ip„a;)]/„(y) as x ^ cx), (4.19)
n=I
and
II
X^(A^ - A^) exp(-2ip„a) exp(-ij9„a;) /„(y) as x ^ -oo.
(4.20)
n=I
It may be noted that A*'" = 0.5/?*'° for n = I, II, where R^j'"" and i?jf are related to
the reflection/transmission coefficients in SM and IM respectively. The expression for re-
spective reflection and transmission coefficients in SM and IM (Manam and Sahoo (2005))
are given below.
The reflection and transmission coefficients in the SM are given by
Kr-i = -\iR} + R'}) exp{-2ipja)
Ktr = -
(i?| - i?f) exp(-2ij9ja)
(4.21)
The reflection and transmission coefficients in the IM are given by
Krn = 2 i^h + ^h) exp(-2ip,,a)
Ktjj =
(i?|j - i?fj) exp(-2ipjja) . (4.22
On the solid boundaries, Eq. 3.15 must be satisfied. Hence the condition on wetted
draft of the dike is given by
dx
— = 0, for < y < d.
(4.23)
The condition across the gap Eq. 3.19 must be imposed to determine the unknowns
{A^, A'^,n = I, II, 1, ...}and{i?^, B"^, n = 0, I, 1, ...}. Hence the continuity of pres-
sure and horizontal velocity at a; = a, give
h = 02 ,
for d < y < H,
(4.24)
4.2. MODEL IN THE CASE OF A SINGLE DIKE 65
^ for d<y < H. (4.25)
dx dx
In case of a two-layer fluid, the eigenfunctions do not foUow the usual orthonormal
relation as in the case of a single-layer fluid. In the present study a general orthonormal
relation suitable for a two-layer fluid is used for both the open water and dike covered
region eigenfunctions.
The eigenfunctions /^'s are integrable in < y < H having a single discontinuity
at y = h, and are orthonormal (/^.'s are normalized eigenfunctions hence the orthogonal
relation becomes orthonormal) with respect to the inner product as given below (see
Manam and Sahoo (2005)).
rh pH
< fn, fm >i= s fn fm dy+ /„ /^ dy, (4.26)
Jo Jh
Similar to /m's, Xn's are also orthonormal with respect to the inner product
< Xn, Xm >2= <
i-h i-H
s XuXm dy+ Xn Xm dy, for d < h,
Jd Jh
(4.27)
H
Xn Xm. dy, for d>h.
d
4.2.3 General Solution Procedure
Using the expressions (4.4) and (4.9) in the Eq. (4.24) and exploiting the orthonormality
of the eigenfunctions Xm's as deflned in relation (4.27), we obtain
1 // oo
B-l=-Y.Cnm+ E KCnm. TU = 0, I , II , 1, . .. , (4.28)
^ n=I n=I,II,l
where, Cnm =< fn, Xm >2- Using expressions (4.4) and (4.9) in Eq. (4.25) and exploiting
the orthonormality of the eigenfunctions /^'s as deflned in relation (4.26), we obtain
-ill oo
-lY^Pn^nm " iPm-4^ = E ^'nln C mn taU q^a, TU = I , II, 1, ..., (4.29)
^ n=I n=0,/,l
where 6nm is the Kronecker delta. Substituting B^ from Eq. (4.28) in (4.29) yields
oo 1 ^^
'^PmA-l + E <nK = ^ T^i^Pr^mr " <^) , m = I, II, 1, ..., (4.30)
n=IJIA ^ r=I
66 CHAPTER 4. WAVE SCATTERING BY SURFACE-PIERCING DIKES
where
oo
«m,^ = Z^ CmrCnrqr tail QrU. (4.31)
mn / J
r=I,l
A similar matching procedure on x = a for the anti-symmetric velocity potential 0"
gives
oo 1 ^^
il'-^m - E («mn + 7mnX = " E(i?''-<^-- + 7m. + «L), "^ = ^, ^^, 1, -, (4-32)
n=IJI,l ^ r=I
where
(^'Ln = Yl ^^r CnrQr COt Q^tt, (4.33)
r=/,l
and
Iran = • (4.34)
a
The systems of equations (4.30) and (4.32) are solved using Gauss-elimination method
to obtain the various physical quantities of interest. These are found to have excellent
convergence characteristics. In the computation number of evanescent modes in the series
are selected based on the experience of the numerical convergence experiment. In numer-
ical convergence experiment a study is carried out to estimate the number of evanescent
modes A^ needed for convergence of the system of equation in the present wave structure
interaction problems. It is observed that 8-10 evanescent modes are enough to obtain the
results accurately up to 3-decimal point in most of the physical situation considered in the
present study. A case study is plotted for wave past porous plate breakwater discussed in
Fig. 7.2 (a and b) of Chapter 7. For computation of all numerical results 15 evanescent
modes are considered.
4.3. MODEL IN THE CASE OF A PAIR OF IDENTICAL DIKES
67
4.3 MODEL IN THE CASE OF A PAIR OF IDEN-
TICAL DIKES
4.3.1 Definition of the Physical Problem
In this section, solution for scattering of incident wave trains by a pair of identical surface-
piercing dikes in a two-layer fluid is considered. The wave scattering by a pair of identical
surface-piercing dikes is shown in Fig. 4.2 is solved, which are again of width 2a and draft
d. The origin (0, 0) is chosen to be at the center of the right-hand dike.
x=-a. x=0 x=a.
y=0
\ \\\ \ \ \ \\
x=-(a+b)
\ \\\ \ \ \ \\
X
y=d
(iii)
(i)
y=h
(ii)
<i
y=H
\\\\\\\\\\\\\\\^
1 \ \ ^
Figure 4.2: Definition sketch for a pair of identical surface-piercing dikes.
4.3.2 Velocity Potentials
The symmetry of the problem is again exploited by writing the solution as the sum of
a symmetric and an anti-symmetric parts. The line of symmetry for present problem
is a; = — (a + 6) (see Fig. 4.2). Hence similar to a single surface-piercing dike we can
write ((){x, y) as a sum of symmetric and anti-symmetric parts as in Eq. (4.1). With
this decomposition of velocity potential, the boundary value problem reduces to a simpler
problem in the region x > —{a + b) only. In addition this decomposition of velocity
potentials helps in increasing the computational efficiency by reducing the number of
unknown coefficients.
68 CHAPTER 4. WAVE SCATTERING BY SURFACE-PIERCING DIKES
The eigenfunction-expansion method similar to the one used in the case of a single
surface-piercing dike is adopted in the present problem also to construct the velocity po-
tentials. There are now two open water regions (region (i) and (in)) and two matching
boundaries. The appropriate eigenfunction-expansions for the symmetric velocity poten-
tials in each region are:
ill oo
(j)'l = -J2 (^wi-iPnix - a))f^{y) + J2 A'^expiip^ix - a))f^{y), (4.35)
^ n=I n=I,II,l
and
,. V^ ^. cosp^(x + a + &)
03 = Z^ D^ : fn{y). 4.36
n=i,ii,i sm p^b
The appropriate eigenfunction-expansions for the anti-symmetric velocity potentials
in each region are:
ill oo
(j)i = -Yl exp(-ip„(a; - a))f^{y) + ^ yl° exp (ip„ (x - a))f^{y), (4.37)
"^ n=I n=I,II,l
and
^_ f O f'^ll^'b ^ly). (4.38)
n=I,II,l ^^^ i^n"
The appropriate eigenfunction-expansions for the symmetric velocity potentials in dike
covered region (region (ii)) is given by
0^ = [B^, + -Q]xo(y) + E [i?:^^^^ + C:^^^^^]x.(y). (4.39)
The appropriate eigenfunction-expansions for the anti-symmetric velocity potentials
in dike covered region (region {ii)) is given by
02 = [Bo + -Qxo(y) + E [Bf^^^ + C:^^^^^]x„(y), (4.40)
a n~I I COS 9ji^ si^ 9»^^
The vertical eigenfunctions and dispersion relations in open water, and dike covered
regions are the same as derived in the case of a single surface-piercing dike. The radiation
condition, definition of refiection and transmission coefficients, condition on the wetted
draft of the dike, continuity condition across the gap are also similar to the case of a single
surface-piercing dike. Moreover, the orthonormality conditions as defined in the case of a
single surface-piercing dike is also applicable in the present problem.
4.3. MODEL IN THE CASE OF A PAIR OF IDENTICAL DIKES 69
4.3.3 General Solution Procedure
The matching procedure for x = ±a is very similar to that used for the single surface-
piercing dike case. The resulting equations for the unknown coefficients are
oo
P™(i^™ - D'J- J2 («mn + 7mn)(^n " K COt pj) =
n=I,II,l
1 ''
-I](iPr<^mr+7mr + «D, rU = I , II, 1,..., (4.41)
r=I
and
oo 1 ^^
n=I,II,l ^ r=I
m = I, II, 1, ..., (4.42)
where a^^ and j^nn are the same as defined in Eqs. (4.31, 4.33 and 4.34). The remaining
coefficients, for the velocity potential 02 (4.39), are given by
ill oo
2i?:. = ;t E C.™ + E Cr.m{K + D:^ cot p^b), m = 0,1,1,..., (4.43)
^ n=I n=I,II,l
and
1 // oo
2C^:^ = ;t E ^-- + E C^UA:, -D:^ cot p^b), m = 0,1,1,.... (4.44)
^ n=I n=I,II,l
where C„^ =< /„, Xm >2-
The final equations for the anti-symmetric velocity potential coefficients {A'^, D°; n =
I, II, 1, ...} and {B^, C°; n = 0, I, 1, ...} are similar in form to Eqs. (4.41) - (4.44)
except that cot p^b must be replaced by tan p^b throughout. This gives
oo
Pmi^Al - D':;^) - E {al^ + lmn){Al-D'^t^npJ) =
n=I,II,l
1 ^^
-E(il'r<^mr + 7mr + aL)> ^ = I , II, 1,..., (4.45)
r=I
and
oo 1 ^^
p™(iA^ + D^)+ E <J^: + ^:tanp„6) = -E(ipA
n=IJI,l ^ r=I
m = I, II, 1, ... . (4.46)
70 CHAPTER 4. WAVE SCATTERING BY SURFACE-PIERCING DIKES
The remaining coefficients, for the vefocity potential 02 (Eq. 4.40), are given by
-ill oo
25™ = ;TECn™+ E C^miAl + D:^ts.npJ), m = 0, /, 1, ..., (4.47)
^ n=I n=I,II,l
and
2C:.= :TECn™+ E C„^«-I?:tanp„6), m = 0, /, 1, ... . (4.48)
^ n=I n=I,II,l
The equation sets (4.41) and (4.42) are solved simultaneously for unknown coefficients
{A'n, D'^, n = I, II, 1, ...}. The equation sets (4.45) and (4.46) are solved simultaneously
to obtain the unknowns {A'^, D*^, n = I, II, 1, ...}. Again Gauss-elimination method
is used to solve the matrix systems and evanescent modes in the series are selected based
on the experience of the numerical convergence experiment.
4.4 MODEL USING WSAM
Finding the solution for more than two dikes, or for multi dikes of different geometry, by an
extension of the previously described methods is a non-trivial task. However, for a given
characteristics of single dikes in isolation, it is possible to obtain approximate solutions for
multi body problems using the WSAM. The distance between dikes is assumed to be large
enough as compared to the wavelength of incident wave trains to neglect local effects while
considering the interactions between the dikes. Hence, the only interactions will result
from the plane waves propagating between the dikes. The results derived using WSAM
for the case of two dikes and the extension to a large number of dikes is straightforward.
The solution procedure in the present section is similar to that of Mclver (1986), which
was described in Srokosz and Evans (1979).
4.4.1 Solution Procedure Using WSAM
Consider a wave incident from large positive x upon two dikes centered on the points
X = Ci and x = C2, Ci > C2. Far from the bodies, the spatial velocity potential ((){x, y) can
4.4. MODEL USING WSAM 71
be written as
//
01 = X! (6^P( ~ ^Pn^) + ^" exp(ij3„a;))/„(y), for x > Ci, (4.49)
n=I
II
(t)2 = J2{^riQW{-^Pnx) + Bnexp{ipnx)^fn{y), for C2 < X < ci, (4.50)
n=I
and
1/
03 = X! ^nexp( - ip^x) /„(y), for x < C2. (4.51)
where R^ and T„ for n = I, II, are related with the reflection and transmission coefficients
for the dike pair. A^ and B^ for n = I, II, are the amplitude of the wave in central
region propagating away from dike 1 and 2 respectively.
The total reflected wave results from the reflection of the incident wave by dike 1 and
the transmission of the second component of 02 through dike 1, hence
Rra = Rrai ©xp ( - 2ip^Ci) + T^iB^, for m = I, II. (4.52)
where Rmn and T^^, for m = I, II are related with the reflection and transmission
coefficients of dike n when in isolation, as defined in the case of a single surface-piercing
dike. Following similar arguments
Am = Trai + RraiBm exp(2ip„Ci), for TU = I , II, (4.53)
-Bm = Rm2Am exp( - 2ip^C2), for m = I, II, (4.54)
and
Tm = Tm2Am, for m = I, II. (4.55)
From Eqs. (4.52) — (4.55), the explicit expressions for R^ and T^ in terms of Rmn and
Tmn, for m = I, II and n = 1, 2 are obtained.
72 CHAPTER 4. WAVE SCATTERING BY SURFACE-PIERCING DIKES
4.5 NUMERICAL RESULTS AND DISCUSSION
Numerical results are computed and analyzed for surface- and internal-wave scattering
by a single and a pair of identical dikes in a two-layer fluid. The effects of various non-
dimensional physical parameters on wave reflection (in both SM and IM) and hydrody-
namic forces experienced by the dikes are analyzed. For convenience, the wave parameters
are given in terms of the non-dimensional wave number pjd, gap between the dikes pjb,
depth ratio h/H, fluid density ratio s along with the non-dimensional dike parameters
given by a/d, H/d and b/H.
4.5.1 Reflected Energy
Here we consider wave reflection by a single and a pair of identical surface-piercing dikes
in both SM and IM. Results are plotted by allowing p^d (normalized wave number in
IM) to vary based on the two-layer fluid dispersion relation.
In Fig. 4.3, present results for single surface-piercing dike reflection coefficients {Kri,
Krii as defined in Eqs. (4.21) and (4.22)) in a two-layer fiuid for different values of H/d
ratios are compared with the results obtained by Mei and Black (1969) in a single-layer
fluid. In general, the reflection coefficients in SM, Kri are similar to that observed by Mei
and Black (1969) except in case of intermediate frequency range, where Kri is found to be
significantly small. It may be noted that in case of a two-layer fiuid, due to the presence
of the interface, waves in SM and IM propagate below the dike (when d < h), which is
not the case for single-layer fiuid. For both small (corresponds to long wave region and
the interface is very close to the dike) and large values of pjd (corresponds to short wave
region and the interface is far from the free surface) the wave transmission in SM due
to interface becomes insignificant. On the other hand, for intermediate frequency range,
the waves in SM transmitted significantly due to the presence of the interface and this
may be the reason for smaller refiection coefficients in SM observed in the present study
as compared to the wave refiection in the single-layer fiuid. When the surface obstacle is
above the interface, the general trend of the IM wave refiection observed in the two-layer
4.5. NUMERICAL RESULTS AND DISCUSSION 73
fluid is found to be similar to the one observed for a bottom obstacle case in a single-layer
fluid (see Fig. 2 of Mei and Black (1969)). However, when the surface obstacle touches
the interface or it extends beyond the interface (IPSO) the reflection in IM is found to be
100 %.
The variation of single dike reflection coefficients in SM and IM versus pjd are plotted
in Fig. 4.4 (a) and (b) respectively for different values of H/d. In Fig. 4.4 (a), for all
values of H/d, with an increase in pjd, the wave reflection in SM increases and attains
a 100 % reflection in the deep water region. In case of intermediate water depth, wave
reflection in SM attains minimum for H/d = 5 and maximum for H/d = 1.5. On the
other hand, for d/H < h/H, the general trend of reflection coefficient in IM follows an
oscillating pattern and it attains a zero reflection in the deep water region (Fig. 4.4 (b)).
This is due to the fact that in the deep water region, the dike is far from the interface
and hence it has a negligible impact on waves in IM. However, as the dike approaches
toward the interface, the wave reflection in IM increases sharply and attains a 100 %
reflection over the entire frequency range in case of IPSO {d > h). This is because in such
situation the propagation of internal waves through the interface as well as free surface is
completely blocked by the rigid dike.
The effect of dike width to mean wetted draft ratio a/d on single dike reflection
coefficients in SM and IM for the single dike are shown in Fig. 4.5 (a) and (b) respectively.
It is observed that with an increase in a/d ratio, the reflection in both SM and IM increases.
This is expected because an increase in dike width will enhance the wave reflection and
when the width becomes inflnitely large, the wave reflection in SM become 100 % over
entire frequency range because in such situation there will not be any transmitted wave
in SM. On the other hand the wave reflection in IM is similar to the one observed in the
case of a bottom obstacle in a single-layer fluid (see Fig. 2 of Mei and Black (1969)).
The effect of depth ratio h/H on the single dike reffection coefficients in SM and IM
for the single dike are shown in Fig. 4.6 (a) and (b) respectively. In Fig. 4.6 (a), it is
observed that the reffection coefficients in SM for h/H = 0.1 and h/H = 0.25 are almost
same except a little change in the intermediate frequency range. Similarly, except in the
74 CHAPTER 4. WAVE SCATTERING BY SURFACE-PIERCING DIKES
shallow water region the difference in the values of wave reflection in SM is marginal for
h/H = 0.5, h/H = 0.75 and h/H = 0.9. On the other hand, the reflection coefficient in
IM is found to be increasing with a decrease in h/H ratio. It is obvious because as the
interface approaches toward the bottom of the dike, the influence of dike on wave motion
in IM is higher, which ultimately leads to a higher wave reflection. When the interface
touches the bottom of the dike {h/H = 0.25) or it is above the dike bottom (the case of
IPSO, h/H = 0.1), wave reflection in IM becomes 100 % and this observation is similar
to the one observed in Fig. 4.4 (b).
The single dike reflection coefficients versus pjd are plotted in SM and IM for various
values of s in Fig. 4.7 (a) and (b) respectively. In general it is observed that wave reflection
in SM increases with an increase in the value of s and a reverse trend is observed in case
of IM wave reflection.
In Fig. 4.8, results for the reflection coefficients in SM and IM versus pjd are plotted for
different values of b/H in case of a pair of identical dikes. A comparison is made between
the results obtained by the matched-eigenfunction-expansion method and the WSAM. It
is observed that WSAM results for reflection coefficients match closely when the dikes
are widely spaced. However, the agreement is closer in the case of wave reflection in IM.
From both the methods in a narrow bandwidth of frequency, the reflection coefficients in
SM reduce suddenly to a very small values. It is observed that with an increase in the gap
between the dikes, more number of these narrow bandwidths of frequency corresponding
to small wave reflection will appear.
4.5.2 Hydrodynamic Forces
It is very interesting to analyze the vertical and horizontal forces experienced by the
surface-piercing dikes as in general they don't have a very strong bottom foundation. The
deflnitions of horizontal force HF and vertical force VF are similar to the one explained
in Mclver (1986) and is given by:
rd
HF = icjp(t)o {(f){a,y) -(f){-a,y))dy, (4.56)
4.5. NUMERICAL RESULTS AND DISCUSSION 75
and
VF = iup(po (p{x,d)dx. (4.57)
J —a
It is clear that symmetric velocity potential does not contribute to the horizontal
wave force. Similar definitions can be obtained for the horizontal and vertical forces,
HF^s and V"F„'s respectively (n = 1 corresponds to the first dike which is exposed to the
incident waves, and n = 2 corresponds to the second dike) in the case of a pair of identical
dikes (see, Mclver (1986)). The behavior of horizontal forces, HF^s and vertical forces,
VF^s per unit incident wave amplitude and length of dikes is analyzed in the subsequent
paragraphs.
The behavior of horizontal and vertical hydro dynamic forces experienced by a single
dike is presented in Fig. 4.9 and 4.10. Resonance behavior is observed in the case of
horizontal hydrodynamic force, which is similar to the one explained in Mclver (1986).
However, it is observed that the magnitude of resonating horizontal hydrodynamic force
increases with a decrease in H/d value and an increase in a/d value. On the other hand,
the vertical hydrodynamic force increases with an increase in pja and in the deep water
region the magnitude of vertical hydrodynamic force is high for small H/d and a/d values.
The variation of horizontal forces, HFJs and vertical forces, VF^s for n = 1, 2 in
the case of a pair of identical dikes is studied for different b/H values in Fig. 4.11 and
4.12 respectively. Similar to Mclver (1986) resonance behaviors are observed in both the
cases of horizontal and vertical hydrodynamic forces. However, it is observed that the
magnitude of resonating horizontal hydrodynamic force is higher on the second dike as
compared to that on the first dike. On the other hand, the magnitudes of resonating
vertical hydrodynamic forces on both the dikes are of same order. Moreover, in the case
of vertical hydrodynamic force the number of resonating peaks are found to be high for
smaller value of b/H, whilst the magnitude of vertical hydrodynamic force at resonance
is found to be high for larger value of b/H.
76 CHAPTER 4. WAVE SCATTERING BY SURFACE-PIERCING DIKES
4.5.3 Summary of Important Observations
The important observations from the present numerical results for surface-piercing dikes
are summarized point wise as below:
1. Present two-layer fluid results for reflection coefficients in SM are found to be similar
to that obtained for single-layer fluid by Mei and Black (1969).
2. When the surface obstacle is above the interface, the general trend of the IM wave
reflection is found to be similar to the one obtained for a bottom obstacle case in a
single-layer fluid by Mei and Black (1969).
3. When dike is placed nearer to the interface, the wave reflection in IM increases
sharply and attains a 100 % reflection over the entire frequency range in case of
IPSO {d > h).
4. With an increase in a/d ratio, the reflection in both SM and IM increases.
5. The reflection coefficient in IM increases with decrease in h/ H ratio. When the
interface touches the bottom of the dike or it is above the dike bottom wave reflection
in IM becomes 100 %.
6. The wave reflection in SM increases with an increases in the value of s and a reverse
trend is observed in case of IM wave reflection.
7. In case of a pair of identical dikes, WSAM results for wave reflection match closely
with the results obtained by matched-eigenfunction-expansion method when the
dikes are widely spaced. The agreement is closer in the case of wave reflection in
IM.
8. In case of a pair of identical dikes, in a narrow bandwidth of frequency, the reflec-
tion coefficients in SM reduce suddenly to very small values. With an increase in
the gap between the dikes more number of these narrow bandwidths of frequency
corresponding to small wave reflection will appear.
9. Similar to Mclver (1986) resonance behavior is observed in case of horizontal hy-
drodynamic forces experienced by a single dike and the magnitude of resonating
horizontal hydrodynamic force increases with a decrease in H /d value and an in-
crease in a/d value.
4.5. NUMERICAL RESULTS AND DISCUSSION
77
10. The vertical hydro dynamic force experienced by a single dike increases with an
increase in pia and in deep water region the magnitude of vertical hydro dynamic
force is high for small H/d and a/d values.
11. Similar to Mclver (1986) resonance behaviors are observed in both cases of horizontal
and vertical hydrodynamic forces for a pair of identical dikes.
12. As compared to first dike the magnitude of resonating horizontal hydrodynamic
force on the second dike is found to be higher.
13. The magnitude of resonating vertical hydrodynamic forces on both the dikes are of
same order.
14. In case of vertical hydrodynamic force the number of resonating peaks are high for
small b/H values, whilst the magnitude of vertical hydrodynamic force at resonance
is found to be high for large values of b/H.
1.2
1
0.8
0.6
0.4
0.2
1 1
1 1 1 1
A
□ ^ — :r:r^^ ^ -
." Mei and Black (1969) Kr I °
/ Present Kr J jj
-'A
'Ia° ' Kin
^ 1 ^ J -
H/d = 2
Mei and Black (1969) Kr ^
Present Krj jj
0.2 0.4
0.6 0.8
p_d
1.2 1.4
Figure 4.3: Comparison of reflection coefficients in SM, Kri and IM, Kru versus pjd for
a single surface-piercing dike at different H/d values, a/d = 1.0, h/H = 0.25 and s = 0.75
with Mei and Black (1969).
78
CHAPTER 4. WAVE SCATTERING BY SURFACE-PIERCING DIKES
Kr,
1
0.8
0.6
0.4
0.2
-
1 1 1
/ / />
■c-
-
1 / /' /
H/d =
H/d =
50 —
10 --■
! /' i
H/d =
5
i
i :'
'Ja'i
//'■•■
H/d =
H/d =
2.5--
1.5 --
- .•//
/ '■■ .•■
_
/<*//•
f
1 1 1
1
1
0.2 0.4 0.6 0.8 1 1.2 1.4
p^d
(a)
Kr„0.6 -;ii;
H/d=50 --
H/d=10 -■
H/d=5 -
H/d = 4.05 ■■
H/d = 4.001 -
H/d = 4- -
p^d
(b)
1.2 1.4
Figure 4.4: Reflection coeflicients in (a) SM, Kri and (b) IM, Kru versus "pid for a single
surface-piercing dike at different if/d values, a/d = 1.0, /i/iJ = 0.25 and s = 0.75.
4.5. NUMERICAL RESULTS AND DISCUSSION
79
1.2
0.8
Ktj 0.6
0.4
0.2
-
1 1 1
1 1
^
__ ^ — -" ^ ^ ^
y
-'a/d^
:0.01--
/
y
^
a/d =
:0.1 -- y
A /
/ /
y
/ W.
'■••. .•■ /
y'
/ /
/a/d =
--\ -
y
y
y'
-»../ /
/ a/d =
1 3 -
y
y'
lll/^
-.^Z a/d =
^5 -
/■' U -^
T' ^
, ^ •
~
1/ /
. ^ ■ -* ■
/• /
^.-■^
/
1 ^ 1-
— f —
1 1
0.2 0.4 0.6 0.8 1
p_d
1.2 1.4
(a)
Kr
II
0.5
0.4
0.3 i
0.2
0.1
1 1 1 1 1
; ; 1
-
1 ' \
1, 1 a/d = 0.01 --
-
I'N; a/d = 0.1 — ■
mk a/d=l
"
^hlh a/d = 3
-
h :;■ a/d = 5 —
-
-
0.2 0.4
0.6 0.8
p_d
1.2 1.4
(b)
Figure 4.5: Reflection coeflicients in (a) SM, Kvi and (b) IM, Kth versus "pid for a single
surface-piercing dike at different ajd values, H jd = 6.0, h/H = 0.25 and s = 0.75.
80
CHAPTER 4. WAVE SCATTERING BY SURFACE-PIERCING DIKES
1.2
0.8
KriO.6
0.4
0.2
-
1 1 1 1 1
.■■;;:^^^^^^^^'''''''''''^ ...r^'-i^^
•y '<^'X '^
/ y- "
-
/
.^^^ h/H = 0.1 —
y
/
J' h/H = 0.25
/
.^
/•-'' h/H = 0.5 --■
- / .^'
/t ''
h/H = 0.75--
/^■y
/'.>'
h/H = 0.9 --
/^' ,
1 1 1 1 1
0.2 0.4
0.6 0.8
1.2 1.4
1.2
0.8
Ktjj 0.6
0.4
0.2
s
.-■ L-
(a)
h/H = 0.1 -
h/H = 0.25
h/H = 0.5 -
h/H = 0.75 -
h/H = 0.9 -
0.2 0.4 0.6 0.8 1
1.2 1.4
(b)
Figure 4.6: Reflection coefficients in (a) SM, Kri and (b) IM, Kth versus "pid for a single
surface-piercing dike at different hjH values, H jd = 5.0, a/d = 1.0 and s = 0.75.
4.5. NUMERICAL RESULTS AND DISCUSSION
1
1 1
- s = 0.1 —
s = 0.25
1 1 1 1
/ .'
.--.
0.8
- s = 0.5 --■
-
s = 0.75 — ■ ,
^
I 0.6
- s = 0.9 --/
.y' /, ^^^
-
0.4
-
0.2
- ^/v (/--^^
"
(Si^^^^-^^^^'l 1
1 1 1 1
0.2 0.4
0.6 0.8 1 1.2
1
\'
(a)
Ktjj 0.3
0.2 0.4 0.6 0.8 1
1.2 1.4
(b)
Figure 4.7: Reflection coeflicients in (a) SM, Kvi and (b) IM, Kth versus "pid for a single
surface-piercing dike at different s values, Hjd = 5.0, a/d = 1.0 and h/H = 0.25.
82
CHAPTER 4. WAVE SCATTERING BY SURFACE-PIERCING DIKES
1.2
1
0.8
Krj 0.6
I I
b/H = 0.25, exact
b/H = 0.75,
exact
WSAM
- WSAM
0.2 0.4 0.6 0.8 1
1.4
(a)
Kr,
b/H = 0.25,
exact -
WSAM
b/H = 0.75,
exact
WSAM -
0.6 0.8
pd
1.2 1.4
(b)
Figure 4.8: Reflection coeflicients in (a) SM, Kri and (b) IM, Kru versus pjd for a pair
of identical surface-piercing dikes at different b/H values, H/d = 6.0, a/d = 1.0, s = 0.75
and h/H = 0.25.
4.5. NUMERICAL RESULTS AND DISCUSSION
0.1
0.08
HF
0.06
0.04
0.02
2.2
H/d = 5
H/d = 6
H/d = 10
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0.02
0.015
0.01
VF
0.005
(a)
1 I I I I I r
H/d = 5
H/d = 6
H/d =10
J I I I I I L
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
(b)
Figure 4.9: (a) Horizontal force, HF and (b) Vertical force, VF per unit incident wave
amplitude and length of dike in MN/m^ for a single surface-piercing dike at different H/d
values, a/d = 1.0, s = 0.75 and h/H = 0.25.
84
CHAPTER 4. WAVE SCATTERING BY SURFACE-PIERCING DIKES
HF
0.1
0.08
0.06
0.04
0.02
0.45
4.9
H r
0.5
0.5
ij a/d = 1
!ia/d = 3
^a/d = 5--i
/.■■• i
/.•••••
r
0.2 0.4 0.6 0.8 1
1.2 1.4 1.6
0.02
0.015
VF
0.01
0.005
(a)
0.2 0.4 0.6 0.8 1
1.2 1.4 1.6
(b)
Figure 4.10: (a) Horizontal force, HF and (b) Vertical force, VF per unit incident wave
amplitude and length of dike in MN/m^ for a single surface-piercing dike at different a/d
values, H/d = 6.0, s = 0.75 and h/H = 0.25.
4.5. NUMERICAL RESULTS AND DISCUSSION
P,d
1.4 1.6
(a)
0.2 0.4 0.6 0.8 1
p_d
1.2 1.4 1.6
(b)
Figure 4.11: Horizontal force on first, \HFi/Io\ and second, \HF2/Io\ dike in MN/m^ for
(a) b/H = 0.25 (b) b/H = 0.75, at H/d = 6.0, a/d = 0.1, s = 0.75 and h/H = 0.25.
86
CHAPTER 4. WAVE SCATTERING BY SURFACE-PIERCING DIKES
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
p^d
(a)
(b)
Figure 4.12: Vertical force on first, \VFi/Io\ and second, \VF2/Io\ dike in MN/m^ for (a)
b/H = 0.25 (b) b/H = 0.75, at H/d = 6.0, a/d = 0.1, s = 0.75 and h/H = 0.25.
Chapter 5
WAVE SCATTERING BY
BOTTOM-STANDING DIKES
5.1 INTRODUCTION
Bottom-standing dikes are considered in this chapter. Like the case of surface-piercing
dikes discussed in Chapter 4, in the present chapter also both the cases of a single and
a pair of identical bottom-standing dikes are considered within the context of linearized-
theory of water waves. Moreover, dikes are again assumed as rigid structures for fluid
flow.
The mathematical model for bottom-standing dikes is similar to that considered in the
case of surface-piercing dikes. The major difference is that there is no dike covered region
in the case of bottom standing dikes. The Cartesian co-ordinate system is again chosen
with x— axis in the direction of wave propagation and y— axis in the downward direction.
Similarly, dikes are approximated as cylinders of rectangular geometry and are placed in a
two-layer fluid of flnite depth. The rectangular dikes are completely immersed in the two-
layer fluid. The problem is considered in the two-dimensions under the assumptions that
the dikes have parallel generators and are of sufficient length for normal waves incidence.
In the two-layer fluid, the upper fluid has a free surface (undisturbed free surface located
at y = 0) and the two fluids are separated by a common interface (undisturbed interface
87
88
CHAPTER 5. WAVE SCATTERING BY BOTTOM-STANDING DIKES
located at y = h), each fluid is of inflnite horizontal extent occupying the region — oo <
2;<cx);0<y</iin case of the upper fluid of density pi, and —oo<x<oo;h<y<H
in case of the lower fluid of density p2- The flow is assumed to be irrotational and simple
harmonic in time with angular frequency oj. Hence as deflned earlier the velocity potential
^{x,y,t) exists such that ^{x,y,t) = Re[((){x,y)exp{—iujt)].
5.2 MODEL IN THE CASE OF A SINGLE DIKE
5.2.1 Definition of the Physical Problem
In the present subsection, solution of wave scattering by a single bottom-standing dike
is analyzed in a two-layer fluid which is further generalized to study the wave scattering
by a pair of identical bottom-standing dikes in the subsequent section. Fig. 5.1 illustrate
the problem under consideration. The dike is of width 2a and draft d. The origin (0, 0)
is chosen to be the point, where the mean free surface intersect with the vertical axis
passing through the center of the dike.
y=0 x=-a x=0 x=a x
y=h
y=H-d
IWWWW
y=H
\
\
\
\
(ii)
(i)
\
\
\
\
\
It
_L
P2
Figure 5.1: Deflnition sketch for single bottom-standing dike.
5.2.2 Velocity Potentials
Following a similar procedure as in the case of surface-piercing dike, in the present case
also the solution is written as the sum of a symmetric and an anti-symmetric parts (see
5.2. MODEL IN THE CASE OF A SINGLE DIKE
i9
Eq. 4.1). The line of symmetry is the line x = 0, which passes through the center of
the dike (see Fig. 5.1). With this decomposition of velocity potential, the boundary
value problem reduces to a simpler problem in the region a; > only. The eigenfunction-
expansion method similar to the one used in the case of a single surface-piercing dike is
adopted in the present problem also to construct the velocity potentials. The symmetric
and anti-symmetric parts of the velocity potential in the region (i) are same as that
described in case of a surface-piecing dike (see Eqs. 4.4 and 4.5). The eigenvalues j3„'s in
the region (i) satisfy the same dispersion relation in p as given in Eq. 4.8. The symmetric
and anti-symmetric parts of the velocity potential in region shallow (ii) are as given below.
u=i,iiA COS q^a
(5.1)
and
= T. B.
aSm qr,x
Xn{y)-
(5.2)
Xn{y) =
n=i,ii,i sm q^a
The vertical eigenfunctions Xn(y)'S) satisfying governing equation (Eq. 3.1) along with
the conditions Eqs. 3.12 — 3.14 and Eq. 3.15 at the top of the dike {y = H — d) are given
by
' L-^sinh q^{H - d- h) [q^ cosh q^y - K sinh q^y]
K cosh q^h - g.sinh q^h ' ^^ ^ < ?/ < ^'
(n = /, //, 1, 2, 3, ...)
L~"^cosh qn{H — d — y), for h < y < {H — d),
(5.3)
where
Ll = (4g„)"^(Kcosh q^h - g„sinh g„/i)"^[(Kcosh q^h - g„sinh g„/i)^
{2{H — d — h)qn + sinh 2qn{H — d — h)) — s sinh^ qn{H — d — h)
iK\2qr,h - sinh 2q^h) - ql{2q^h + sinh 2q^h) + 2Kq^{cosh 2q^h - 1))]. (5.4)
Similar to Eq. 4.8 the eigenvalues q^s satisfy the dispersion relation in q as given by
(1 — s)q^ tanh q{H — d — h) tanh qh — gKftanh qh + tanh q{H — d — h)]
-K'^[s tanh q{H - d - h) tanh qh + I] = 0. (5.5)
90 CHAPTER 5. WAVE SCATTERING BY BOTTOM-STANDING DIKES
It may be noted that in region (ii), in case of (H — d) > h, the vertical eigenfunctions
have two propagating modes because of the presence of the free surface and the interface.
The vertical eigenfunctions in region (ii) are similar with the vertical eigenfunctions in the
region (i). On the other hand, in case of {H — d) < h (special case of bottom obstacle), the
vertical eigenfunctions x^'s do not satisfy the interface conditions Eqs. (3.13 and 3.14)
and in such situation like a single-layer fluid, only one propagative mode exists because of
the free surface. Hence, similar to single-layer fluid, the term n = II does not appear in
the expression of eigenfunctions and the corresponding vertical eigenfunctions becomes
cosh qn{H — d)
where
^2 _ '2qn{H - d) + sinh 2g„(iJ" - d)
4 g^cosh qn{II - d)
and Qn satisfy the relation
K = Qn tanh QniH — d), for n = I, 1, 2, ... (5-8)
The radiation condition, definition of refiection and transmission coefficients, condition
on the wetted draft of the dike, continuity condition across the gap are similar to the case
of a single surface-piercing dike.
The eigenfunctions /^'s are integrable in < y < H having a single discontinuity at
y = h, and are orthonormal with respect to the inner product as given in Eq. 4.26 for the
case of surface-piercing dikes.
Similar to /m's, Xn^ sue also orthonormal with respect to the inner product
rh rH-d
s XnXmdy+ Xn Xm dy, for {H - d) > h,
Jo Jh
(5.9)
pH-d
/ Xn Xm dy, for {H - d) < h.
^ Jo
< Xn, Xm >3= <
5.2.3 General Solution Procedure
The matching procedure at x = a for symmetric velocity potentials in the case of bottom-
standing dike is very similar to that used in the case of surface-piercing dike. The resulting
5.3. MODEL IN THE CASE OF A PAIR OF IDENTICAL DIKES 91
equation is given by
oo 1 ^^
iPm^m + E <nAn = l^J^i^PrSmr " <J, m = I, II, 1, ... . (5.10)
n=I,II,l ^ r=I
The matching procedure at x = a for anti-symmetric velocity potentials in the case of
bottom-standing dike is also very similar to that used in the case of surface-piercing dike.
The resulting equation is given by
oo 1 ^^
'^PmAl- Yl <n<=:^E(iM™^ + <.)' m = I, 11,1, ..., (5.11)
n=I,II,l ^ r=I
where the expressions for a'^ are same as defined in Eqs. (4.31 and 4.33).
The systems of equations (5.10) and (5.11) are solved using Gauss-elimination method
to obtain the various physical quantities of interest. Number of evanescent modes in the
series are selected based on the experience of the numerical convergence experiment.
5.3 MODEL IN THE CASE OF A PAIR OF IDEN-
TICAL DIKES
5.3.1 Definition of the Physical Problem
In the present section, solution for scattering of incident wave trains by a pair of identical
bottom-standing dikes in a two-layer fiuid is considered. The wave scattering by a pair
of identical bottom-standing dikes, as shown in Fig. 5.2 is solved. Dikes are again of
width 2a and draft d. The origin (0, 0) is chosen to be at the center of dike located on
the right-hand of Fig. 5.2.
5.3.2 Velocity Potentials
The symetricity of the problem is again exploited by writing the solution as the sum of
a symmetric and an anti-symmetric parts. The line of symmetry for present problem is
X = —{a + b) (see Fig. 5.2). Hence similar to single surface-piercing dike we can write
92
CHAPTER 5. WAVE SCATTERING BY BOTTOM-STANDING DIKES
y=0
x=-(a+b) x=-a (0,0) x=a
(iii) JM
\\\\\\\\\
\
<
\
\
\
\
\
\
\
\
\
\
k y\
f k
(i)
?
Figure 5.2: Definition sketch for a pair of identical bottom-standing dikes.
velocity potential (()(x, y) as a sum of symmetric and anti-symmetric parts as in Eq. (4.1).
With this decomposition of velocity potential, the boundary value problem reduces to a
simpler problem in the region x > —{a + b) only. The eigenfunction-expansion method
similar to the one used in the case of a single surface-piercing dike is adopted in the
present problem also to construct the velocity potentials.
Similar to the case of a pair of identical surface-piercing dikes there are now two regions
which are beyond the dike width (region (i) and (in)) and two matching boundaries. The
symmetric and anti-symmetric parts of the velocity potential in the region (i) are same
as that presented in case of a pair of identical surface-piecing dikes (see Eqs. 4.35 and
4.37). The symmetric part of the velocity potential in region (iii) is as given below.
03= E K ^ 7 -fn{y).
n=i,ii,i sm p^b
The anti-symmetric part of the velocity potential in region (iii) is as given below.
(5.12)
n=I,II,l
COS Pnb
(5.13)
The symmetric part of the velocity potential in shallow region (ii) is as given below.
CXJ
02= E [b]
,cos QnX „^sm q^xi
. + C„- \xn{y)-
n=I,II,l
COS q^a
sm q^a
(5.14)
The anti-symmetric part of the velocity potential in region (ii) is given below.
^2= E [b:
,cos q^x sm q^Xi
: + C„- \xn{y)-
n=I,II,l
COS q^a
sm q^a
(5.15)
5.3. MODEL IN THE CASE OF A PAIR OF IDENTICAL DIKES 93
The vertical eigenfunctions and dispersion relations in regions beyond the dike width
and in region over the dikes are same as described in the case of a single bottom-standing
dike. The radiation condition, definition of refiection and transmission coefficients, con-
dition on the wetted draft of the dike, continuity condition across the gap are also similar
to the case of a single surface-piercing dike. Moreover, the orthonormality conditions as
described in the case of a single bottom-standing dike is also applicable in the present
problem.
5.3.3 General Solution Procedure
Matching procedure for x = a is very similar to that used for the single bottom-standing
dike case. The resulting equations for the unknown coefficients are
oo 1 ^^
PmiiA'^ -D'J- J2 <^SK - K cot Pnh) = - J2i^Pr5mr + «L),
n=I,II,l ^ r=I
m = I, II, 1, ..., (5.16)
and
1 ''
P^{iAl + D'J+ Yl <niAn + K cot Pnb) = -J2i^PrSmr-atnr),
n=I,II,l ^ r=I
m = I, II, 1, ..., (5.17)
where a^^ are the same as defined in Eqs. (4.31 and 4.33). The remaining coefficients,
for the velocity potential 02 (Eq. 5.14), are given by
1 // oo
'^B-:^ = i;T.Cnm+ E C^miK + K^Otp^b), TU = 1 , 1 1 , 1, . .. , (5.18)
^ n=I n=I,II,l
and
ill oo
2C^ = ;T E Cnr. + E CnmiA:, - D^ COt p^b) , TU = I , II, 1, ... . (5.19)
^ n=I n=I,II,l
where C^m =< fn, Xm >3-
The final equations for the anti-symmetric velocity potential coefficients {A^, D°; n =
I, II, 1, ...} and {B^, C^; n = 0, /, 1, ...} are similar in form to Eqs. 5.16 — 5.19
94 CHAPTER 5. WAVE SCATTERING BY BOTTOM-STANDING DIKES
except that cot Pnb must be replace by tan Pnb throughout. This gives
oo 1 ^^
n=IJIA ^ r=I
m = I, II, 1 ,..., (5.20)
and
oo 1 ^^
P^(iA^ + D^)+ J2 <niK + K^^^Pnb) = -J2i^Pr5mr-a'^r),
n=I,II,l ^ r=I
m = I, II, 1, ... . (5.21)
The remaining coefficients, for the velocity potential 02 (Eq. 5.15), are given by
ill oo
2i?™ = ;TE^™+ E C„™(^: + I?:tanp„6), m = I, II, I, ..., (5.22)
^ n=I n=I,II,l
and
-\ II oo
'^Cl = -Y.Cr.^+ E C^ra{Al-Dlts.np^h), TU = 1 , 1 1 , 1, . .. . (5.23)
^ n=I n=I,II,l
The equation sets (5.16) and (5.17) are solved simultaneously for unknown coefficients
{A^, D'^, n = I, II, 1, ...}. The equation sets (5.20) and (5.21) are solved simultaneously
to obtain the unknowns {A°, DJJ, n = I, II, 1, ...}. Again Gauss-elimination method
is used to solve the matrix systems and evanescent modes in the series are selected based
on the experience of the numerical convergence experiment.
All the equations (Eqs. 4.49 — 4.55) derived applying WSAM for the case of a pair of
identical surface-piercing dikes in section (4.5) is also valid for the present case where a
pair of bottom-standing dikes are considered in a two-layer fluid.
5.4 NUMERICAL RESULTS AND DISCUSSION
Numerical results are computed and analyzed for surface- and internal- wave scattering by
a single and a pair of identical bottom-standing dikes in a two-layer fluid. The effects of
various non-dimensional physical parameters on wave reflection in both SM and IM are
analyzed. For convenience, the wave parameters are given in terms of the non-dimensional
wave number pid, gap between the dikes pih, depth ratio h/ H , fluid density ratio s along
with the non-dimensional dike parameters given by a/d, H/d and b/H.
5.4. NUMERICAL RESULTS AND DISCUSSION 95
5.4.1 Reflected Energy
The variation of reflection coefficients in SM and IM versus pi{H — d) are plotted in
Fig. 5.3 (a) and (b) respectively for different values of H /d. In general, it is observed
that the wave reflection in both SM and IM are found to be increasing with a decrease
in H /d. When the bottom-standing dike is in the lower fluid domain {H — d > h, i.e
H/d = 2.0 and 3.0), with an increase in pi{H — d), the wave reflection in both SM and IM
decreases and approaches to zero in the deep water region. This is because the bottom-
standing dike has a negligible impact on the wave motion in both the modes in the deep
water region. However, when the dike is extended up to the upper fluid (in case of IPBO,
H — d < h, i.e H/d = 1.05 and 1.15), the wave reflection in SM is significantly high and
attains a maximum value in the intermediate frequency range (Fig. 5.3 (a)). For IPBO,
the wave refiection in IM is found to be increasing with an increase in pi{H — d) and the
trend suggests that the refiection in IM attains 100 % refiection in the deep water region
(Fig. 5.3 (b)). This is because in such situation the interface is far from the free surface
and the waves in IM cannot get transmitted by the free surface and this will lead to a
situation where there will be no wave transmission in IM.
The effect of dike width to mean wetted draft ratio a/d on reffection coefficients in
SM and IM are shown in Fig. 5.4 (a) and (b) respectively. It is observed that with an
increase in a/d ratio, the refiection in both SM and IM increases. The general trend of
the refiection coefficients in both SM and IM is similar to the one observed in the case of
a bottom-standing dike in a single-layer fiuid (see Fig. 2 of Mei and Black (1969)).
The effect of interface location h/H on reffection coefficients in SM and IM are shown
in Fig. 5.5 (a) and (b) respectively. In general it is observed that with a decrease in h/H
ratio the refiection in SM increases (Fig. 5.5 (a)). On the other hand, the wave refiection
in IM increases with an increase in the value of h/H (Fig. 5.5 (b)). However, the general
trend of IPBO case refiection coefficients in both SM and IM is different than that of
non-IPBO case.
Reffection coefficients are plotted versus pi{H — d) in SM and IM for various values
of s in Fig. 5.6 (a) and (b) respectively. It is observed that the wave refiection in SM has
96 CHAPTER 5. WAVE SCATTERING BY BOTTOM-STANDING DIKES
higher reflection peaks for higher values of s and a reverse trend is observed in case of IM
wave reflection.
In Fig. 5.7 (a) and (b), results for the reflection coefficients in SM and IM versus pid
are plotted for different values of b/H in case of a pair of identical dikes. A comparison is
made between the results obtained by the matched-eigenfunction-expansion method and
the WSAM. Similar to case of pair of surface-piercing dikes, it is observed that WSAM
results for reflection coefficients match closely when the dikes are widely spaced.
5.4.2 Summary of Important Observations
The important observations from the present numerical results for bottom-standing dikes
are summarized point wise as below:
1. Wave reflection in both SM and IM are increasing with a decrease in H/d.
2. When the bottom-standing dike is in the lower fluid domain {H — d > h), with
an increase in pi{H — d), the wave reflection in both SM and IM decreases and
approaches to zero in the deep water region.
3. When the dike is extended up to the upper fluid (IPBO, H — d < h), the wave re-
flection in SM is signiflcantly high and attains a maximum value in the intermediate
frequency range.
4. For IPBO, the wave reflection in IM is increasing with an increase in pi{H — d)
and the trend suggests that the reflection in IM attains 100 % reflection in the deep
water region.
5. With increase in a/d ratio, the reflection in both SM and IM increases. The general
trend of the reflection coefficients in both SM and IM is similar to the one observed
in the case of a bottom-standing dike in a single-layer fluid (Fig. 2 of Mei and Black
(1969)).
6. With a decrease in h/H ratio the wave reflection in SM increases and the wave
reflection in IM decrease.
7. Wave reflection in SM has higher reflection peaks for higher values of s and a reverse
trend is observed in case of IM wave reflection.
5.4. NUMERICAL RESULTS AND DISCUSSION 97
8. Similar to case of pair of surface-piercing dikes, reflection coefficients obtained by
WSAM and matched-eigenfunction-expansion method for bottom-standing dikes
match closely when the dikes are widely spaced.
98
CHAPTER 5. WAVE SCATTERING BY BOTTOM-STANDING DIKES
KTj 0.4
0.3
0.2
^
H/d=1.15
H/d = 2.0 -
H/d = 3.0 -
N
I V ^ 1 - L
0.2 0.4 0.6 0.8 1
Pj (H-d)
1.2 1.4 1.6
Kr,
(a)
H/d=1.05
H/d=1.15
H/d=2.0
H/d=3.0
Pj (H-d)
(b)
1.6
Figure 5.3: Reflection coeflicients in (a) SM, Kri and (b) IM, Kru versus pi{H — d) for a
single bottom-standing dike at different H /d values, a/d = 6.0, h/H = 0.25 and s = 0.75.
5.4. NUMERICAL RESULTS AND DISCUSSION
99
Kr,
0.35
0.3
0.25
0.2
0.15
l.l ''
\l'.
/ I >;.!■■■■■■•■•/.
I '-.A
i/ '/i' :'
I- .1
ni ':!>^ ^rjix
"1 I I
a/d = 0.1 —
a/d = 1.0
a/d = 2.0 - -
a/d = 4.0 — ■
a/d = 6.0 --
0.2 0.4 0.6 0.8 1
Pj (H-d)
Kr
II
0.25
0.2
0.15
0.1
0.05
\ r.
Mm
'-' /M\
(a)
\
-.•70 -!->./
ii/ ! \i. i../^-\
H/v I. ^. i ^' ^V\
"1 I I
a/d = 0.1 -
a/d =1.0 ■■
a/d = 2.0 -
a/d = 4.0 -
a/d = 6.0 -
iL_^\/ ^-^M.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
p/H-d)
(b)
Figure 5.4: Reflection coeflicients in (a) SM, Krj and (b) IM, Krjj versus pi{H — d) for a
single bottom-standing dike at different a/d values, H /d = 2.0, h/H = 0.25 and s = 0.75.
100
CHAPTER 5. WAVE SCATTERING BY BOTTOM-STANDING DIKES
Kr,
0.7
0.6
0.5
0.4
0.3
0.2
0.1
N r^ / \
N , I
"1 I I I I I r
h/H = 0.1
^ I '
h/H = 0.25
h/H = 0.5 --
h/H = 0.75 — ■
h/H = 0.9 --
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
p/H-d)
(a)
1.2
0.8
Kfjj 0.6
0.4
0.2
"1 I I I I r
-.-■Ji'-^.-. ...I-
h/H = 0.1 —
h/H = 0.25
h/H = 0.5 --■
h/H = 0.75 — ■
h/H = 0.9 --
_L I I I
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
p/H-d)
(b)
Figure 5.5: Reflection coeflicients in (a) SM, Krj and (b) IM, Krn versus pi{H — d) for a
single bottom-standing dike at different h/H values, H /d = 2.0, a/d = 6.0 and s = 0.75.
5.4. NUMERICAL RESULTS AND DISCUSSION
101
Kr,
0.2 0.4 0.6 0.8 1
Pj (H-d)
(a)
Kr 0.15
0.2 0.4 0.6 0.8 1
Pj (H-d)
1.2 1.4 1.6
1.2 1.4 1.6
(b)
Figure 5.6: Reflection coeflicients in (a) SM, Krj and (b) IM, Krjj versus pi{H — d) for a
single bottom-standing dike at different s values, H /d = 2.0, a/d = 6.0 and h/H = 0.25.
102
CHAPTER 5. WAVE SCATTERING BY BOTTOM-STANDING DIKES
1.2
1
0.8
KTj 0.6
0.4
0.2
i..i
a I, n^
t; nil; M 1 {
»; !!; : i h ::
»i!!;ii;A(,fi ;!,,ii
b/H = 0.25
exact — WSAM
b/H = 0.75
exact - - WSAM
II
i...»<tci>^:?-^s)^.i
1.2 1.4 1.6
Pj (H-d)
(a)
Krjj 0.3
1 I
b/H = 0.25
exact — WSAM ■■■■
b/H = 0.75
exact - - WSAM -
0.2 0.3 0.4 0.5 0.6 0.7 0.8
p/H-d)
(b)
Figure 5.7: Reflection coeflicients in (a) SM, Kri and (b) IM, Kru versus pjd for a pair
of identical surface-piercing dikes at different b/H values, H/d = 6.0, a/d = 1.0, s = 0.75
and h/H = 0.25.
Chapter 6
WAVE PAST POROUS
MEMBRANE BREAKWATER
6.1 INTRODUCTION
After analyzing the studies on rigid rectangular surface-piercing and bottom-standing
dikes, wave scattering by flexible porous breakwaters are considered. Wave past flexible
porous structures involves relatively complex flow physics and difficult mathematical for-
mulation. In the present chapter, scattering of water waves by a flexible porous membrane
breakwater in a two-layer fluid having a free surface is analyzed. Similar to the problems
considered in the past and in the sequel, the present problem is also formulated based on
the usual 2D potential flow assumption with time harmonic potential.
6.2 DEFINITION OF THE PHYSICAL PROBLEM
In wave scattering by flexible porous membrane in a two-layer fluid (Fig. 6.1), the fluids
are separated by a common interface (undisturbed surface located at y = /i). The upper
fluid has a free surface (undisturbed surface located at y = 0), and each fluid is of inflnite
horizontal extent (— oo < x < +oo); both the upper and lower fluid are of flnite depth,
< y < h and h < y < H respectively. Region 1 is deflned as— cx)<2;<0,0<y<iJ
103
104
CHAPTER 6. WAVE PAST POROUS MEMBRANE BREAKWATER
and region 2 is defined as < x < +oo; < y < H (see Fig. 6.1). The porous membrane
is located Sitx = 0,0<y<H.
Incident wave in
surface mode
Porous membrane breakwater
^^^^
(0,0) /
Free surface
X
Incident wave in
internal mode
►-
':
Q)
Interface
y^ii
; y = H
Figure 6.1: Definition sketch for fiexible porous membrane breakwater.
6.3 MODEL FOR FLUID FLOW
Considering the waves incident from large negative x upon the fiexible porous membrane,
the velocity potentials are obtained by eigenfunction-expansion method, similar to the
case of surface-piercing dike, in each of the two regions 1 and 2 as marked in Fig. 6.1.
The present fiuid fiow problem is a boundary value problem which satisfies the fiuid fiow
governing equation (3.1) along with the conditions Eqs. 3.12 — 3.14 and Eqs. 3.16 —
3.18. The velocity potentials in the regions 1 and 2 are given by
II
n=I
lp„x
J2 Rne ^''"'')fn{Pn,y), fora;<0,
1=1,11,1
(6.1)
and
oo
^2= E TjP''''.UPn,y), forx>0.
n=I,II,l
(6.2)
6.4. MODEL FOR MEMBRANE RESPONSE
105
The eigenfunctions /„'s are given by
' sinh Pn{H - h) [p^ cosh p^y - K sinh p^y
fn{Pn, y)
K cosh Pnh — Pn sinh Pnh
for < y < h,
= <
{n = I, II, 1, 2,...) (6.3)
cosh Pn{II — y), for h < y < H,
where, Rn and T„ for n = I, II, 1, 2, 3, ... are unknown constants to be determined.
Note that in the present problem only open water regions exist. Hence, the wave
numbers p^ {n = I, II for positive real roots and n = 1, 2, 3, ... for positive purely
imaginary roots) are the roots of the dispersion relation in p as given in Eq. 4.8.
Similar to the case of single surface-piercing dikes, in the present problem the open
water region eigenfunctions /„'s for n = I, II, 1, 2, 3, ... are integrable in < y < H
having a single discontinuity at y = h and are orthogonal (Manam and Sahoo (2005))
with respect to the inner product as defined by
rh r-H
< fn, fm >4= S / fnfmdy + / fnfmdy.
Jo Jh
(6.4)
The reflection and transmission coefficients in SM and IM are defined by (Manam and
Sahoo (2005))
Kri =
Krii =
h
Rii
hi
and Kti =
and Ktji =
Tj_
h
Tn
In
in SM,
in IM.
(6.5)
6.4 MODEL FOR MEMBRANE RESPONSE
As described earlier in the general mathematical formulation, it is assumed that break-
water is defiected horizontally with displacement C{y,t) = Re[^{y)e~^'^^], where {(y) rep-
resents the complex defiection amplitude and is assumed to be small as compared to the
water depth. It is assumed that the membrane is a thin, homogeneous and inextensible
sheet with uniform mass nig {nis = Psb, b is the thickness of the membrane, Ps is the
uniform membrane mass density) under constant tension T. With these assumptions, the
governing equation (Eq. 3.21) relating the membrane displacement ( from equilibrium to
106 CHAPTER 6. WAVE PAST POROUS MEMBRANE BREAKWATER
that of differential pressure acting on the membrane at x = can be appHed. This gives
Z'^fHi^' " (6.6)
T
d^^ 2 _ I -Y-i^2 - 0i), for < y < /i,
^^ ^"1 ^^^2-0i), for/i<y<i/,
where P = uJuig/T is the breakwater frequency parameter.
The membrane is pinned at the free surface and at the bottom. The corresponding
boundary conditions as given in Eq. 3.24 can be applied. This gives
e(0) = 0, e(^) = 0. (6.7)
The continuity condition across the interface (Eq. 3.30) should be imposed. Hence
the continuity of deflection and slope of the membrane breakwater across the interface
(the point on the breakwater where the two fluid meet each other (x = 0; y = h)) yield
{(/.-) = {(/i+), i\h-) = ah^)- (6.8)
Applying the porous boundary condition (Eq. 3.36) on the porous membrane break-
water, we obtain
' = ikoG{(^i - 02) + iuji (j = 1, 2) on X = 0, < y < H, (6.9)
dx
where G is the complex porous-effect parameter as defined in Eq. 3.37.
6.5 GENERAL SOLUTION PROCEDURE
Applying the continuity of (j)^ (Eq. 6.9) along the porous breakwater on a; = and
invoking the orthogonality relation (Eq. 6.4) over {0 < y < h) Li {h < y < H), we obtain
In — Rn = Tn for u = I, II and R^ = —T^ for n = 1, 2, 3, ... . (6.10)
Utilizing the Eqs. 6.1, 6.2 and 6.10 a general solution is obtained for the second order
non-homogeneous ODE, Eq. 6.6 (membrane breakwater governing equation) and is given
by
ay) = C'e^'' + C"e-^'y-'^ f -^Up^,y) hTO<y<H, (6.11)
6.5. GENERAL SOLUTION PROCEDURE
107
where the arbitrary constants C, C" and the fluid density p are defined as below.
Ci for < y < /i, C2 for < y < /i, pi for < y < h,
C' = i C" = l p = l (6.12)
C3 for h < y < H, C4 for /i < y < i/, \ p2 for h < y < H.
Substituting the general solution for { (Eq. 6.11) in Eq. 6.9 and using the relations
in Eqs. 6.1, 6.2 and 6.10 the following expression is derived.
hoiy) - J2 Rnhuiy) = 0, 0<y<H,
n=I,II,l
(6.13)
where
ho{y) =
Wilifiivi, y) + miliifiiiPii, y) - iojCi e^'^^ - iujC2 e ^^^, for < y < /i,
^ iPiIifiipi, y) + iPiiIiifiiipii, y) - iwC's e^'^^ - iujC^ e"^^^, for h <y < H,
(6.14)
and
hniy) =
\ , l.rj. + '^Prr + 2ip/G] fnipn, y), for < y < /i,
{n = I, II, 1, 2,...)
(6.15)
2oj'^P2
[\pl + P')T
ipn + 2ipjG] fn{Pn, y) , ioi h < y < H ,
We can apply the least-squares-approximation method to Eq. 6.13 as described earlier
in general mathematical formulation chapter. We can write
N
Q{y) = ho{y) - J2 Rnhn{y), for < y < i/.
n=I,II,l
Applying the least-squares method, we obtain
H _ dO(v)
Qiy)^^dy = 0, hTn = I, II, 1, 2,...,N,
(6.16)
(6.17)
where the bar denotes the complex conjugate.
Eq. 6.17 provides A^ + 2 linear equations with A + 6 unknowns, as /io(y) involves
4 extra unknowns Ci, C2, C3 and C4. Substituting the expression for { (Eq. 6.11) in
the end conditions on the breakwater as in Eq. 6.7 and the continuity conditions at the
108 CHAPTER 6. WAVE PAST POROUS MEMBRANE BREAKWATER
interface as in Eq. 6.8 yield the required another 4 hnear equations. These system of
equations are solved using Gauss-elimination method to compute and analyze various
physical quantities of interest. Number of evanescent modes in the series are selected
based on the experience of the numerical convergence experiment.
6.6 NUMERICAL RESULTS AND DISCUSSION
In the present section, numerical results on the combined effect of porosity and membrane
tension are discussed, to analyze the performance of membrane breakwater in the two-
layer fluid for various non-dimensional parameters. The wave and membrane parameters
are given in terms of non-dimensional values of wave number piH, water depth h/H,
fluid density ratio s, porous-effect parameter G, membrane tension T' = T/(pig/i^), and
membrane mass m' = nis/pih. The membrane mass m' is kept fixed {m' = 0.1) through-
out the analysis as the effect of membrane mass on the performance characteristic of the
breakwater is insignificant (Kim and Kee (1996), and Lo (1998)).
6.6.1 Reflected and Transmitted Energy
The wave transmission across a permeable fiexible breakwater is governed by two com-
bined phenomena. When a train of waves approaches a permeable fiexible breakwater,
seepage fiow induced by waves penetrates through the breakwater and waves are repro-
duced with some dissipation after transmission. On the other hand, due to deformation
of the fiexible breakwater, the waves are regenerated in the downstream side, even if there
is no fiow across the breakwater.
In general, the energy refiection and transmission provide one of the major criteria in
deciding the efiectiveness of the breakwater. In this subsection, the effect of various non-
dimensional physical parameters on energy reffection and transmission in both SM and IM
are analyzed. For the sake of simplicity, all results in the present subsection are analyzed
with respect to the normalized SM wave number pjH by allowing the normalized IM wave
number pnH to vary based on the two-layer fluid dispersion relation. It is observed from
6.6. NUMERICAL RESULTS AND DISCUSSION 109
the general trend of wave reflection in SM that the wave reflection decreases from its peak
to a certain value in the shallow water region and thereafter it attains a constant value.
On the other hand, the wave reflection in IM increases from zero to a certain value in
the shallow water region and thereafter it attains a constant value (see Figs. 6.2 — 6.5).
Similar results are obtained for wave reflection by a flexible membrane breakwater in a
single-layer fluid by Lo (2000) and Lee and Lo (2002) (see Fig. 3 (a) of Lo (2000) and Fig.
5 of Lee and Lo (2002)). Furthermore, the wave reflection in SM is found to be significantly
smaller than the wave refiection in IM, which suggests that a membrane breakwater is
more effective in IM wave motion than in SM wave motion. Similar observations for
porous breakwaters in a two-layer fluid are reported by Manam and Sahoo (2005).
In Fig. 6.2 (a) and (b), the reflection and transmission coefficients in SM and IM
respectively are plotted against piH, for different values of membrane tension parameter
T'. It is observed that higher wave transmission and lower wave reflection occur in SM
where as lower wave transmission and higher wave reflection occur in IM over the range
of practical interest.
The variation of reflection and transmission coefficients versus pjH for both SM and
IM are plotted in Fig. 6.3 (a) and (b) respectively for different values of the porous-effect
parameter G. In general, the wave reffection in both SM and IM increases with a decrease
in the value of |G| and a reverse trend is observed in the case of wave transmission. This
is expected, because an increase in porosity not only allows more waves to pass through
the breakwater but also reduces the membrane breakwater resistance to the wave motion.
The effect of non-dimensional water depth h/H of two ffuids on the reffection and
transmission coefficients in SM and IM are shown in Fig. 6.4 (a) and (b) respectively. In
SM wave motion it is observed that the wave transmission is lower and the wave reflection
is higher for a thinner upper layer i.e., for h/H = 0.25 (Fig. 6.4 (a)). However, except
for very small values of pjH the wave reflection and transmission are same for h/H = 0.5
and 0.75. On the other hand, an opposite trend is observed in case of IM wave motion
where the wave transmission is higher and the wave reflection is lower for a thinner upper
layer i.e., h/H = 0.25 (Fig. 6.4 (b)) almost over the entire range of interest. This may be
no CHAPTER 6. WAVE PAST POROUS MEMBRANE BREAKWATER
due to the resonating interaction induced by vertical flexible breakwater, between surface-
and internal-waves, when the free surface is close to the interface.
The reflection and transmission coefficients versus pjH are plotted in SM and IM for
different fluid density ratio s in Fig. 6.5 (a) and (b) respectively. In Fig. 6.5 (a) it is
observed that the fluid density ratio s has negligible effect on both wave reffection and
transmission for SM wave motion. However, the wave reffection in SM is observed to be
marginally higher for large value of s {s = 0.75). On the other hand, the wave reffection
increases and the wave transmission decreases for IM wave motion with an increase in
ffuid density ratio (Fig. 6.5 (b)). This nature of the wave transmission in IM may be due
to the high interface elevation as the ffuid density ratio s approaches to unity (Kundu
and Cohen (2002) and Milne-Thomson (1996)).
6.6.2 Free Surface and Interface Elevations
The nature of free surface elevation rjfs and interface elevation rjmt versus non-dimensional
distance x/ Xj are studied after normalizing with respect to the amplitude of the incident
waves in the surface mode. This normalization gives a clear understanding about the
amplitude of the free surface elevation to that of interfacial wave elevation. The free
surface and interface elevations near the breakwater are the result of mutual interaction
of propagating and evanescent modes of both surface and internal-waves (see Figs. 6.6 —
6.8). Hence the free surface and interface elevations in a two-layer ffuid are combinations
of two prominent wave patterns which are referred to as primary and secondary wave
patterns in the present paper. The primary pattern is the one which is generated due to
SM wave motion and the secondary wave pattern is that developed due to the IM wave
motion. In general, it is observed that the interface elevation is much larger than that of
the free surface elevation when either the densities of the two ffuids are very close or in the
case when the interface and free surface are close to each other. A similar situation exists
in a real ocean, as explained theoretically in Milne-Thomson (1996) (see page 445). One
of the reasons for such a high wave amplitude may be due to the resonating interaction
between the waves in SM and IM.
6.6. NUMERICAL RESULTS AND DISCUSSION 111
Fig. 6.6 (a) and (b) show the pattern of the free surface and interface elevation
respectively for different values of membrane tension parameter T'. The effect of change
in tension T' is significant only near the locations of local maxima and minima of the
secondary wave pattern in the case of free surface elevation Fig. 6.6 (a). On the other
hand, the interface elevation is found to be independent of the variation in membrane
tension (see Fig. 6.6 (b)).
Variation of free surface and interface elevation at different h/H ratios are shown in
Fig. 6.7 (a) and (b) respectively. It is observed that as the interface and free surface be-
come nearer, the amplitudes of both free surface and interface elevations becomes higher.
This may be due to the resonating interaction between the waves in SM and IM. The
magnitude of the primary and secondary wave pattern amplitudes of the free surface ele-
vation are of same order for small h/H ratio (Fig. 6.7 (a)). This is due to the fact that
the interface elevation increases rapidly when free surface and interface are close to each
other (see Fig. 6.7 (b)).
Fig. 6.8 (a) and (b) show the pattern of the free surface and interface elevations
for different fluid density ratios s. It is observed that the amplitude of the free surface
elevation increases with decrease in the fluid density ratio s (Fig. 6.8 (a)). On the other
hand, an opposite trend is observed in case of interface elevation where amplitudes of
the interface increases with an increase in fluid density ratio. As the fluid density ratio s
approaches one, the secondary wave pattern of the free surface and the interface elevations
amplify rapidly, which is a well known phenomenon in the case of inter-facial waves (see
Kundu and Cohen (2002) and Milne-Thomson (1996)). It is important to note that among
the elevations, the interface depends heavily on the density ratio s. The reason for this
is that the amplitudes of waves in IM are very sensitive to the change in density ratio s
whereas the waves in SM are least affected by the change in the value of s. Hence interface
elevations change sharply with the change in parameter s whereas free surface elevations
are comparatively less affected by the change in the value of s. The variation in free
surface elevations with the change in s is mainly due to the existence of the secondary
wave pattern, which is again caused by the internal- waves. This is the reason why, in
112 CHAPTER 6. WAVE PAST POROUS MEMBRANE BREAKWATER
Fig. 6.5, the reflection and transmission coefiicients in IM are more dependent on s than
those in SM. Interestingly, when s = 0.25, the free surface elevation is free from secondary
waves as in this case the internal-waves have very small amplitude. Furthermore, it is
observed that with increase in the value of s the wave length of interfacial waves reduces
and very short waves are observed as s approaches one.
The local effects are not visible in the elevation plots because magnitude of the con-
tribution of local effects is insignificant as compared to that of propagating modes in SM
and IM in the present study. Moreover, their contribution decays quickly as one moves
away from the breakwater (either left or right) because of the exponential decay of the
multiplication factor in the velocity potential. There is always a discontinuity in elevation
as the waves pass the breakwater. However, in the present case the magnitude of the
discontinuity is very small because the porous membrane offers very little resistance to
waves. The discontinuity is only apparent in Fig. 6.8 (a) for s = 0.25.
6.6.3 Response of Membrane Breakwater
In the present subsection, the variation of membrane breakwater response { normalized
with respect to incident wave amplitude // in SM is analyzed for various membrane and
two-layer ffuid parameters. In all of Figs. 6.9 — 6.12, the vanishing nature of membrane
response at the two ends is because the membrane is fixed at those points.
Variation of normalized membrane response \^/Ii\ for different h/H ratios is plotted
versus normalized vertical position y/H in Fig. 6.9. It is observed that the breakwater
has higher deffection amplitude at a location nearer to the interface. This is due to the
propagation of surface and interfacial waves at the interface in a two-layer ffuid. However,
the deflection is found to be higher for small h/H ratio (the interface is closer to the free
surface). This is because of the higher free surface and interface elevation as observed in
Fig. 6.7.
Variation of the normalized membrane response |{///| is plotted versus normalized
vertical position y/H for different values of ffuid density ratio s in Fig. 6.10. The mem-
brane deffection is found to increase with the increase in ffuid density ratio s. The reasons
6.6. NUMERICAL RESULTS AND DISCUSSION 113
for these observations are clear from the nature of free surface and interface elevations in
Fig. 6.8 (a) and (b). However nearer to the free surface the membrane deflection in the
upper fluid domain is found to be high for low fluid density ratio s = 0.25 as in this case
the amplitude of free surface elevation is found to be quite high (see Fig. 6.8 (a)).
The normalized membrane response |{///| for various values of membrane tension
parameter T' is plotted versus normalized vertical position y/H in Fig. 6.11. It is clear
from Fig. 6. 11 that the membrane deflection increases with decrease in membrane tension.
This is expected, because a reduction in membrane tension leads to a reduction in the
stiffness of the membrane against the wave motion and leads to a higher membrane
deflection.
In Fig. 6.12 the normalized membrane response |{///| is plotted versus normalized
vertical position y/H for different values of the porous-effect parameter G. A high mem-
brane deflection is observed for higher values of the imaginary part of the porous-effect
parameter G (the inertia effect of the ffuid inside the porous breakwater).
6.6.4 Summary of Important Observations
The important observations from the present numerical results for porous membrane are
summarized pointwise as below:
1. General trend of wave reffection in SM decreases from its peak to a certain value
in the shallow water region and thereafter it attains a constant value. The wave
reffection in IM increases from zero to a certain value in the shallow water region
and thereafter it attains a constant value. Similar results are reported for wave
reffection by a ffexible membrane breakwater in a single-layer ffuid (see, Fig. 3 (a)
of Lo (2000) and Fig. 5 of Lee and Lo (2002)).
2. Wave reffection in SM is found to be signiffcantly smaller than the wave reffection
in IM, which suggests that a membrane breakwater is more effective in IM wave
motion than in SM wave motion. Similar observations for porous breakwaters in a
two-layer ffuid are reported by Manam and Sahoo (2005).
114 CHAPTER 6. WAVE PAST POROUS MEMBRANE BREAKWATER
3. In general, wave reflection in both SM and IM increases with a decrease in the value
of porous effect parameter |G| and a reverse trend is observed in the case of wave
transmission.
4. In SM wave transmission is lower and the wave reflection is higher for a thinner
upper layer and an opposite trend is observed in case of IM wave motion.
5. Fluid density ratio s has negligible effect on both wave reffection and transmission for
SM wave motion. The wave reffection increases and the wave transmission decreases
for IM wave motion with an increase in ffuid density ratio.
6. Free surface and interface elevations are combinations of primary and secondary
wave patterns.
7. Interface elevation is much larger than that of the free surface elevation when either
the densities of the two ffuids are very close or in the case when the interface and
free surface are close to each other. A similar situation exists in a real ocean, as
explained theoretically in Milne-Thomson (1996).
8. The effect of change in tension T' is significant only near the locations of local max-
ima and minima of the secondary wave pattern in the case of free surface elevation.
The interface elevation is independent of the variation in membrane tension.
9. As the interface and free surface become nearer, the amplitudes of both free surface
and interface elevations becomes higher.
10. Amplitude of the free surface elevation increases with decrease in the fiuid density
ratio s and an opposite trend is observed in case of interface elevation.
11. As the fiuid density ratio s approaches one, the secondary wave pattern of the
free surface and the interface elevations amplify rapidly, which is a well known
phenomenon in the case of inter-facial waves (see Kundu and Cohen (2002) and
Milne-Thomson (1996)).
12. Among the elevations, the interface depends heavily on the density ratio s.
13. The variation in free surface elevations with the change in s is mainly due to the
existence of the secondary wave pattern, which is caused by the internal- waves.
6.6. NUMERICAL RESULTS AND DISCUSSION 115
14. With increase in the value of s the wave length of interfacial waves reduces and very
short waves are observed as s approaches one.
15. Breakwater has higher deflection amplitude at a location nearer to the interface.
16. Breakwater deflection is higher for small h/H ratio (the interface is closer to the
free surface).
17. Membrane deflection increases with the increase in fluid density ratio s.
18. Nearer to the free surface the membrane deflection in the upper fluid domain is
found to be high for low fluid density ratio s = 0.25 as in this case the amplitude
of free surface elevation is found to be quite high.
19. Membrane deflection increases with decrease in membrane tension.
20. Membrane deflection is higher for high value of imaginary part of the porous-effect
parameter G (the inertia effect of the fluid inside the porous breakwater).
116
CHAPTER 6. WAVE PAST POROUS MEMBRANE BREAKWATER
1
0.8
0.6
0.4
0.2
V /f/
1 1
Kt',
V • / '
~ \ '■ /' •■
U' / T' = 0.4
- V-i r = o.2
-
K \ r = 0A --■
/' ■' Y '•
- // .' \\
-
Kr,
/'.' \^
1 1 1 1 1
P,H
(b)
Figure 6.2: Reflection and transmission coeflicients in (a) SM and (b) IM versus pjH for
different T' values at G = 1 + 2i, s = 0.75 and h/H = 0.5.
6.6. NUMERICAL RESULTS AND DISCUSSION
117
Kt
G = l —
^W G = 2
'/A G = l+2i - -
Kr
P^H
(a)
1
0.8
0.6
0.4
0.2
1 1
^<>'-y^^ "
r ■* ^s. "^
/' ^^^^
. /
Kt„
/ G = 1
-/ G = 2
-
/ G = l+2i - - .
1 1
IjH
(b)
Figure 6.3: Reflection and transmission coefficients in (a) SM and (b) IM versus pjH for
different G values at h/H = 0.5, s = 0.75 and T' = 0.4.
118
CHAPTER 6. WAVE PAST POROUS MEMBRANE BREAKWATER
P^H
(a)
1
0.8
0.6
0.4
0.2
\
\ _
/
;
/
;
I
1 1
1 1 1
\- ^,^<^' ""
}C
y ^<:^
f
Kt„
' 7
h/H = 0.75
1 y
h/H = 0.5
h/H = 0.25 - - ■
1 1
1 1 1
3 4
P,H
(b)
Figure 6.4: Reflection and transmission coeflicients in (a) SM and (b) IM versus piH for
different h/H ratios at G = 1 + 2i, s = 0.75 and T' = 0.4.
6.6. NUMERICAL RESULTS AND DISCUSSION
119
1
0.8
0.6
0.4
0.2
\ \ ■
\ ■
Vv '
|..LJ'U^ 1 -
1
Kt,
-
\h s
= 0.25
= 0.5
-
'■> s
1 \ '■ ^
= 0.75 --■
~ r\ ■ ^
-
/■' \ ■ ^
N.»
Kr,
1 1
1
2 3 4
(a)
1
0.8
0.6
0.4
0.2
■cr^ 1 1 1 1 1
\
Kt„\
"^
/
/
/
-
Kr„/
/ ..■■■•■
1 ..••■'"
.y
-■■' '/
s = 0.25
■■/
s = 0.5
f
1
s = 0.75 --.
,
P,H
(b)
Figure 6.5: Reflection and transmission coefficients in (a) SM and (b) IM versus piH for
different s values at h/H = 0.5, G = 1 + 2i and T' = 0.4.
120
CHAPTER 6. WAVE PAST POROUS MEMBRANE BREAKWATER
Vh
-0.08
-2 -1.5 -1
-0.5 0.5
1 1.5 2
(a)
^int/Ij
(b)
Figure 6.6: (a) Free surface and (b) Interface elevation versus x/Xj for different T' values
at piH = 1.0, h/H = 0.5, G = 1 + 2i and s = 0.75.
6.6. NUMERICAL RESULTS AND DISCUSSION
121
0.8
0.6
0.4
0.2
h/H = 0.25
li/H = 0'.5 ' h/H = 0.75--
\/l^
-0.2
-0.4
-0.6
-0.8
-2 -1.5 -1
n
-0.5 0.5
1 1.5 2
10
llint/I, 1
-5
-10
(a)
h/H = 0.25 ^ h/H = 0.5 ■ h/H = 0.75
-2 -1.5 -1
-0.5 0.5
x/?ij
1 1.5 2
(b)
Figure 6.7: (a) Free surface and (b) Interface elevation versus x/Xj for different h/H
ratios at pjH = 1.0, G = 1 + 2i, s = 0.75 and T' = 0.4.
122
CHAPTER 6. WAVE PAST POROUS MEMBRANE BREAKWATER
^fs/Ii
-0.05
-0.15
J I I I I I L
-2 -1.5 -1
-0.5 0.5
1 1.5 2
(a)
I I I I I I r
s = 0.25 — s = 0.5 s = 0.75
(b)
Figure 6.8: (a) Free surface and (b) Interface elevation versus x/Xj for different s values
at piH = 1.0, h/H = 0.5, G = 1 + 2i and T' = 0.4.
6.6. NUMERICAL RESULTS AND DISCUSSION
123
y/H
0.2
0.4
0.6
h/H = 0.75 —
h/H = 0.5
h/H = 0.25 - -
J I
0.1 0.2 0.3 0.4 0.5 0.6
1^/1,1
Figure 6.9: Membrane displacement versus y/H for different h/H ratios at s = 0.75,
PiH = 1.0, T' = 0.4 and G = 1 + 2i.
y/H
"1 1 1 1 1 r
s = 0.25
s = 0.5
s = 0.75
I \^^- ~\- I I I I I I I L
0.02 0.04 0.06 0.08 0.1
Figure 6.10: Membrane displacement versus y/H for different s values at h/H = 0.5,
PiH = 1.0, T' = 0.4 and G = 1 + 2i.
124
CHAPTER 6. WAVE PAST POROUS MEMBRANE BREAKWATER
y/H
0.2
0.4
0.6
0.8
1
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
Ph I
Figure 6.11: Membrane displacement versus y/H for different T' values at h/H = 0.5,
s = 0.75, piH = 1.0 and G = 1 + 2i.
S. 1 1 1 1 1
1 1 1
T' = 0.4 —
1 ^ ■••■
T' = 0.2 -
s
\ ^
\ ^ *.
\ ^ •
~
~
T' = 0.1 --■
s
y<l--~'~\~ " \ 1 1 1
-
'
1 1 1
y/H
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
Figure 6.12: Membrane displacement versus yjH for different G values at s = 0.75,
ViH = 1.0, h/H = 0.5 and T = 0.4.
Chapter 7
WAVE PAST POROUS PLATE
BREAKWATER
7.1 INTRODUCTION
After analyzing wave past flexible porous membrane breakwater in Chapter 6, wave scat-
tering by flexible porous plate breakwater is considered in the present chapter.
7.2 DEFINITION OF THE PHYSICAL PROBLEM
Fig. 7.1 illustrates physical problem for wave scattering by flexible porous plate in a
two-layer fluid under consideration. Similar to Chapter 6, in this chapter as well, fluids
are separated by a common interface (undisturbed surface located at y = /i), wherein the
upper fluid has a free surface (undisturbed surface located at y = 0), and each fluid is of
inflnite horizontal extent (— oo < x < +oo); both the upper and lower fluid are of flnite
depth, < y < h and h < y < H respectively. Region 1 is deflned as — oo < a; < 0,
< y < H and region 2 is deflned as < a; < +oo; < y < H (see Fig. 7.1). The porous
flexible plate is located Sitx = 0,0<y<H.
125
126
CHAPTER 7. WAVE PAST POROUS PLATE BREAKWATER
Incident wave in
surface mode
Porous plate breakwater
^ — ^
(0,0) /
Free surface
X
Incident wave in
internal mode
►
Q)
'.'
•y
G)
Interface
y = h
; y = H
Figure 7.1: Definition sketch for fiexible porous plate breakwater.
7.3 MODEL FOR FLUID FLOW
Similar to the case of porous membrane breakwater, by considering the waves incident
from large negative x upon the fiexible porous plate, the velocity potentials are obtained
by eigenfunction-expansion method in each of the two regions 1 and 2 as marked in Fig.
7.1. The boundary value problem is defined by equation (3.1) along with the conditions
Eqs. (3.12 — 3.14) and Eqs. 3.16 — 3.18). Applying the eigenfunction-expansion method,
the velocity potentials in the regions 1 and 2 can be obtained and are same as given in
Eqs. 6.1 and 6.2. Moreover the eigenfunctions /„'s are also same as defined in Eq. 6.3.
As only open water regions exist in this case, the wave numbers p^ {n = I, II for
positive real roots and n = 1, 2, 3, ... for positive purely imaginary roots) are the roots
of the dispersion relation in p as given in Eq. 4.8. The orthogonality condition as defined
in the inner product Eq. 6.4 for the case of membrane is also applicable in the present
problem. The refiection and transmission coefficients in SM and IM as defined in the case
of membrane problem in Eq. 6.5 are also same in the present problem.
7.4. MODEL FOR FLEXIBLE PLATE RESPONSE 127
7.4 MODEL FOR FLEXIBLE PLATE RESPONSE
As described earlier in the general mathematical formulation, it is assumed that plate
breakwater is deflected horizontally with displacement C{y,t) = Re[^{y)e~^'^*], where ^{y)
represents the complex deflection amplitude and is assumed to be small as compared to
the water depth. It is assumed that the plate breakwater is thin and behaves like a one-
dimensional beam of uniform flexural rigidity EI and mass per unit length m^. With
these assumptions, the governing equation (Eq. 3.20) relating the plate displacement {
from equilibrium to that of differential pressure acting on the plate at a; = can be
applied. This gives
^ _ a'l = I rT'"*' " ■*''■ ^°' ° *" ^ * '"' (7 n
where (3 is the structural frequency parameter as deflned hy (3 = {m sUj"^ / E ly / '^ . The
breakwater will behave like a cantilever as it is assumed in the study that the breakwater
has free and flxed ends at the free surface and seabed respectively.
The plate breakwater is clamped at the seabed and has a free end at the free surface.
The corresponding boundary conditions as given in Eqs. 3.22 and 3.23 can be applied.
This gives
C"(0) = 0, i"'{Q) = 0, i{H) = 0, i'iH) = 0. (7.2)
The continuity condition across the interface (Eq. 3.29) should be imposed. Hence
the deflection, slope of deflection, bending moment and the shear force acting on the plate
breakwater are continuous at the interface (the point on the breakwater where the two
fluids meet each other (x = 0; y = h)). This yield
i{h-) = i{h^), ^'ih-) = eih^), eih-) = eih^), e'ih-) = e"ih+). (7.3)
Applying the porous boundary condition (Eq. 3.36) on the porous plate breakwater
we obtain the same expression as deflned in Eq. 6.9 and G is the complex porous-effect
parameter as deflned in Eq. 3.37.
128
CHAPTER 7. WAVE PAST POROUS PLATE BREAKWATER
7.5 GENERAL SOLUTION PROCEDURE
Applying the continuity of 0^ (Eq. 6.9) along the porous breakwater on x = and
invoking the orthogonality relation (Eq. 6.4) over {0 < y < h) Li {h < y < H), we can
obtain the expression as defined in Eq. 6.10.
Utilizing the Eqs. 6.1, 6.2 and 6.10 a general solution is obtained for the fourth order
non-homogeneous ODE, Eq. 7.1 (plate breakwater governing equation) and is given by
({y) = Eie'^y + E^e-'^y + E^e''^ + E^e
^i^y _L p.q/'^'
EI ^ v^ + B^
^^ n=I,II,l fn^ I-'
for (0 <y < H),
(7.4)
where the arbitrary constants Ei, i = 1, 2, 3, 4 and the fiuid density p are defined as
below.
Ci for < y < h,
Ei = <
Pi for < y < h,
(i = 1, 2, 3, 4), p = \ (7.5)
P2 for h < y < H.
Di for h < y < H,
Substituting the general solution for { (Eq. 7.4) in Eq. 6.9 and using the relations in
Eqs. 6.1, 6.2 and 6.10 the following expression is derived.
where
ho{y)
= <
ho{y)- E Rnhn{y) = 0, {0<y<H)
n=IJIA
^[pilifiiPi, y) + PiiliifiiiPii, y) - c^(Ci e'^y + C2 e-'^y + C3 e^^+
C4 e"^^)], for < y < /i,
'APih.fi{Pi, y) + PiiiiifiiiPii, y) - ^{Di e'^y + d^ e-'^y + Ds e^y+
(7.6)
hn{y)
= <
2 oj^ pi
{pi - (i')EI
2uj^ P2
['{pi- (3')EI
D4 e"'^^)], ioi h<y<H,
+ ipn + 2ipiG ] fn{Pn, y) , foT < y < h,
{n = I, II, 1, 2, ...)
+ ipn + 2ipiG ] fniPn, y) , ioi h < y < H .
(7.7)
(7.8)
7.6. NUMERICAL RESULTS AND DISCUSSION 129
We can apply the least-squares-approximation method to Eq. 7.6 as described earlier
in general mathematical formulation chapter. We can write
JV
Q{y) = ho{y)- E Rnhn{y), for < y < //. (7.9)
n=I,II,l
Applying the least-squares method, we obtain
r Q{y)^§^dy = 0, forn = /, //, 1, 2,...,N, (7.10)
Jo OKn
where the bar denotes the complex conjugate.
Eq. 7.10 provides A^ + 2 linear equations with A + 10 number of unknowns, as ho{y)
involves 8 extra unknowns C^'s and Di's (for i = 1, 2, 3, 4). Another required 8 linear
equations are obtained from the breakwater end conditions (Eq. 7.2), interface conditions
(Eq. 7.3) and the expression for { in Eq. 7.4. These system of equations are solved using
Gauss-elimination method to compute and analyze various physical quantities of interest.
Number of evanescent modes in the series are selected based on the experience of the
numerical convergence experiment.
7.6 NUMERICAL RESULTS AND DISCUSSION
Numerical results are generated to study the combined effect of porosity and flexibility
of breakwater on the wave motion in a two-layer fluid. The wave parameters are given in
terms of the non-dimensional wave number piH, water depth h/H , fluid density ratio s
and the breakwater parameters like the flexural rigidity EI / p2gH^, porous-effect param-
eter G, and mass per unit length m^. The breakwater mass nig is kept fixed at nig = 10
Kg/m^ throughout the analysis (same numerical value for nig is taken by Wang and Ren
(1993)) because it is observed by Williams and Wang (2003) that the breakwater mass
density has a minimal infiuence on the efficiency of the structure as a barrier to the wave
motion.
A case study for the numerical convergence experiment is plotted in Fig. 7.2 (a and
b), which depict the effect of number of selected evanescent modes, A on the accuracy
of reflection/transmission coefficients in SM and IM respectively for wave past porous
130 CHAPTER 7. WAVE PAST POROUS PLATE BREAKWATER
plate breakwater. It may be seen from Fig. 7.2 (a and b), that for A^ = 10 and 15 the
deviation in results is insignificant. In the present study, 15 evanescent modes are taken
for computation of all numerical results.
7.6.1 Reflected and Transmitted Energy
In this subsection, the effects of various non-dimensional physical parameters on wave
reflection and transmission in both SM and IM are analyzed. All results are presented
with respect to the normalized SM wave number piH by allowing normalized IM wave
number pnH to vary, based on the dispersion relation. The effect of non-dimensional
breakwater flexural rigidity on the reflection and transmission coefficients in SM and IM
are shown in Fig. 7.3 (a) and (b) respectively. The general pattern of wave reflection in
both SM and IM is that it increases from zero to it's peak in the shallow water region
and thereafter it reduces and there will be a negligible reflection in the deep water region.
Similar observation are found for wave reflection by a plate barrier in a single-layer fluid
in past (see. Figs. 4.4 — 4.8 of Williams and Wang (2003)). Wave reflection is increasing
and wave transmission is reducing with the increase in the stiffness of the breakwater for
both SM and IM wave motion. This is intuitively expected, as a stiffer structure will
resist more waves, which as a result leads to higher wave reflection. Similar observations
were made by Wang and Ren (1993) in the analysis of wave scattering by flexible barrier
in a single-layer fluid domain of constant density.
The variation of reflection and transmission coefficients versus pjH in both the cases
of SM and IM are plotted in Fig. 7.4 (a) and (b) respectively for different values of the
porous-effect parameter G. Highest wave reffection and lowest wave transmission peaks
(in both SM and IM cases) are observed for the case where breakwater has zero porosity,
which is similar to the observations in case of a single-layer fluid by Williams and Wang
(2003). However, it is observed that for G having large value of inertia effect of ffuid
inside the porous breakwater, wave reffection (in both SM and IM cases) is high. The
probable reason may be that, in such situation the waves are obstructed signiflcantly by
the porous breakwater.
7.6. NUMERICAL RESULTS AND DISCUSSION 131
The effect of depth ratio h/H of two fluids on the reflection and transmission coeffi-
cients in SM and IM are shown in Fig. 7.5 (a) and (b) respectively. In SM wave motion,
highest wave reflection peak is observed for h/H = 0.25 and lowest wave transmission
peak is observed for h/H = 0.5 (see Fig. 7.5 (a)). However, the general pattern of wave
reflection and transmission in case of SM wave motion is not signiflcantly affected by the
interface location. On the other hand, in IM wave motion, when the interface is located
either very nearer to free surface {h/H = 0.1) or seabed {h/H = 0.9) (see Fig. 7.5 (b)),
wave reflection increases and accordingly there is a reduction in the wave transmission.
The probable reason for this change in general pattern of wave reflection and transmission
in case of IM wave motion is due to the resonating interaction of surface- and internal-
waves for h/H = 0.1 and it is due to influence of seabed for h/H = 0.9. In a two-layer
fluid, the thin upper layer can be found in ocean where upper layer density changes due
to solar heating and the thin lower layer can be found in ocean where near to the seabed
the fluid density changes due to the mud and salinity.
The reflection and transmission coefficients are plotted versus pjH in SM and IM for
various values of s in Fig. 7.6 (a) and (b) respectively. The general pattern of wave
reflection and transmission in case of SM wave motion is not signiflcantly affected by the
change in the value of s (see Fig. 7.6 (a)). However, in case of wave motion in IM, when
ffuid density ratio approaches to 1 (s = 0.99 and 0.995) reffection coefficient increases
initially with an increase in pjH and maintains a uniform value for higher wave number
i.e., in the deep water region (see Fig. 7.6 (b)). In general, it is observed in case of IM
wave motion that wave reflection is increasing and wave transmission is decreasing with
the increase in the value of s. This is intuitively expected, as the fluid density ratio s
approaches to unity, the interface elevation becomes signiflcantly high (see Kundu and
Cohen (2002) and Milne-Thomson (1996)). This increase in the elevation of the interface
helps more waves in IM to reflect. For most of the situation in a real ocean, the fluid
density ratio s is very close to one. This observation suggests that although the reflection
coefficient for wave motion in SM is not significantly affected, the effect of the wave
motion in IM cannot be neglected as s ^ 1. For these kind of situations, the structure
132 CHAPTER 7. WAVE PAST POROUS PLATE BREAKWATER
will experience high wave load due to the impact of internal- waves.
7.6.2 Response of Plate Breakwater
In the present subsection, the breakwater response is analyzed. Unlike the case of reflec-
tion and transmission coefficients, the plate response is computed based on the combined
effect of the waves in SM and IM apart from the local effects.
Variation of breakwater response |{///| for different values of non-dimensional break-
water ffexural rigidity EI / p2gH'^, porous-effect parameter G and depth ratio h/ H are
plotted in Figs. 7.7 — 7.9. The breakwater deffection is increasing with a decrease in the
rigidity of breakwater in Fig. 7.7. This is intuitively expected because less rigid structure
will deform or bend more under the action of wave load. In Fig. 7.8, the breakwater dis-
placement is high for G = 0. Similar observations are made in a single-layer ffuid study
(see, Wang and Ren (1993), Fig. 5). This is because less porous structure will experience
a higher force. However, for very high porosity the strength of the structure will reduce
and may cause higher bending (see the cases of G = 2 and 1 + 2i). The bending of the
breakwater is increasing with decrease in the value of h/ H in Fig. 7.9. This is because,
a cantilever will bend more when the location of the concentration of load is at higher
distance from the fixed end. In addition, as the free surface and interface are close to each
other, the wave load on the structure becomes high near the free surface.
7.6.3 Hydrodynamic Force on Plate Breakwater
In the subsection, the hydrodynamic force on the breakwater is analyzed. Unlike the
case of refiection and transmission coefficients, hydrodynamic force is computed based
on the combined effect of the waves in SM and IM apart from the local effects. The
hydrodynamic force coefficient Kf is given by Kf = \Fo/ p2gHh\, where
Fo = ioof p[02(O,y)-0i(O,y)]dy. (7.11)
Jo
Hydrodynamic force coefficients Kf acting on the breakwater versus non-dimensional
ffexural rigidity EI / p2gH^ for different values of G and h/ H are presented in Figs. 7.10
7.6. NUMERICAL RESULTS AND DISCUSSION 133
and 7.11. In general, the hydro dynamic force increases with an increase in flexural rigid-
ity and then attains constant value for higher values of EI / p2gH^. This is because at
higher value of EI / p2gH^, the breakwater start behaving like a rigid wall. Further, the
hydrodynamic force on the structure reduces with an increase in porosity as expected
(Fig. 7.10). Similar observations were made in case of a single-layer fluid by Wang and
Ren (1993). The wave load on the breakwater is reducing with the increase in the value
of h/ H in Fig. 7.11. With an increase in the value of h/ H , the combined effect of waves
in SM and IM reduces and hence the force on the breakwater reduces.
7.6.4 Summary of Important Observations
The important observations from the present numerical results for porous plate are sum-
marized pointwise as below:
1. The general pattern of wave reflection in both SM and IM is that it increases from
zero to it's peak in the shallow water region and there after it reduces and there will
be a negligible reflection in the deep water region. Similar observations are found
for wave reflection by a plate barrier in a single-layer fluid in past (see. Figs. 4.4 —
4.8 of Wilhams and Wang (2003)).
2. Wave reflection is increasing and wave transmission is reducing with the increase
in the stiffness of the breakwater for both SM and IM wave motion. Similar obser-
vations were made by Wang and Ren (1993) in the analysis of wave scattering by
flexible barrier in a single-layer fluid domain of constant density.
3. Highest wave reflection and lowest wave transmission peaks (in both SM and IM
cases) are observed for the case where breakwater has zero porosity, which is similar
to the observations in case of a single-layer fluid by Williams and Wang (2003).
4. Wave reflection (in both SM and IM cases) is high for G with large value of inertia
effect of fluid inside the porous breakwater.
5. General pattern of wave reflection and transmission in case of SM wave motion is
not signiflcantly affected by the interface location.
134 CHAPTER 7. WAVE PAST POROUS PLATE BREAKWATER
6. In IM wave motion, when the interface is located either very near to free surface or
seabed, wave reflection increases and accordingly there is a reduction in the wave
transmission.
7. The general pattern of wave reflection and transmission in case of SM wave motion
is not significantly affected by the change in the value of s.
8. In general, it is observed in case of IM wave motion that wave reffection is increasing
and wave transmission is decreasing with the increase in the value of s.
9. Breakwater deflection is increasing with the decrease in the rigidity of breakwater.
10. Breakwater displacement is high for G = 0. Similar observations are made in a
single-layer fluid study (see, Wang and Ren (1993), Fig. 5).
11. The bending of the breakwater is increasing with decrease in the value of h/H.
12. Hydrodynamic force increases with an increase in flexural rigidity and then attains
constant value for higher values of EI / p2gH^. Similar observations were made in
case of a single-layer fluid by Wang and Ren (1993).
13. The wave load on the breakwater is reducing with the increase in the value of h/ H .
7.6. NUMERICAL RESULTS AND DISCUSSION
135
(a)
(b)
Figure 7.2: Convergence test for reflection and transmission coeflicients in (a) SM and (b)
IM versus piH in case of wave past porous plate breakwater problem at EI / p2gH'^ = 0.01,
G = 1, s = 0.75 and h/H = 0.25.
136
CHAPTER 7. WAVE PAST POROUS PLATE BREAKWATER
1
0.8
0.6
0.4
0.2
KtT
4
XVL J
EI/pgH = 0.2 -
^ 4 ,
EI/pgH = 0.1
_
2 4
EI/pgH = 0.06 --■
2 4
"f^^
EI/pgH = 0.02 --
EI/pgH^O.Ol - -
N A ^"r~~-^^
2
\ \ -/^^^^
s V •
\"o^. ■■■■O:
^^^
^. ^. -.
* • . . ^""--^sss.^^^
r -"^-S.-.^-F-s;...
.^ — ':^.Ui.^ ■ ,7!7^^-sax
P^H
(a)
1
0.8
0.6
0.4
0.2
Kt
II
EI/p gH = 0.2 —
EI/pgH^=0.1
EI/p grf= 0.06 --
EI/p gH^= 0.02 --
EI/p gH = 0.01 -■-
(b)
Figure 7.3: Reflection and transmission coeflicients in (a) SM and (b) IM versus pjH for
different EI / p2gH^ values at G = 1, s = 0.9 and h/ H = 0.25.
7.6. NUMERICAL RESULTS AND DISCUSSION
137
1
0.8
0.6
0.4
™:.--*r
Kt
G =
G=l
G = l+0.5i
G = 2 --
G=l+2i -
T'^
2 3
(a)
1
0.8
0.6
0.4
0.2
!-JW— T
G = —
G=l
G=l+0.5i
G = 2 --
G = l+2i -
■^-^^TT^fc^T'C^T^^w
2 3 4
1"
(b)
Figure 7.4: Reflection and transmission coeflicients in (a) SM and (b) IM versus pjH for
different G values at h/H = 0.75, s = 0.75 and EI/p2gH^ = 0.02.
138
CHAPTER 7. WAVE PAST POROUS PLATE BREAKWATER
1
0.8
0.6
0.4
h/H = 0.1
h/H = 0.25
h/H = 0.5
h/H = 0.75
h/H = 0.9
(a)
/h/H = 0.1 -
''"^- ^■' h/H = 0.25 ■■
h/H = 0.5 --
h/H = 0.75 -■-■
h/H = 0.9 --
i___.
2 3
(b)
Figure 7.5: Reflection and transmission coeflicients in (a) SM and (b) IM versus pjH for
different h/H ratios at G = 2, s = 0.9 and EI / p2gH^ = 0.1.
7.6. NUMERICAL RESULTS AND DISCUSSION
139
1
0.8
0.6
0.4
0.2
w -r^ -f
\.-
Kt
Kr
s = 0.5
s = 0.75
s = 0.9
s = 0.99
s = 0.995
2 3
(a)
1
0.8
0.6
0.4
0.2
■\ ■■■---.-•■•
■\ / s-0.5 s = 0.99
P,N .' s = 0.75 s = 0.995 -- "
\ \ Ktji s = 0.9 --.
ilk \
f ' \ ^
^ .x^- X ::■•..... __^^__^^^^^^
2 3
1"
(b)
Figure 7.6: Reflection and transmission coeflicients in (a) SM and (b) IM versus pjH for
different s values at h/H = 0.75, G = 1 + 0.5i and EI/p2gH^ = 0.06.
140
CHAPTER 7. WAVE PAST POROUS PLATE BREAKWATER
y/H
0.2
0.4
0.6
0.8
1
1— j r-r
/
/
/
/
/
/
'III
III
'■;i
I
■■'■I ••
'/
EI/pgH = 0.01
EI/pgH^=0.02
2
EI/pgH^=0.06
2
EI/pgH^O.!
EI/pgHto.2
3
l^/I,
Figure 7.7: Breakwater displacement profile for different EI /p2gH'^ vafues at h/H = 0.25,
s = 0.9, piH = 0.5 and G = 1.
y/H
0.2
0.4
0.6
0.8
1
1
1
1
1
\ ■' T 1 1 ^
/
•^ ^ —
/
y" ^^
/
/•■
/
-^ ^^
/ ••' ,r ^
/
■ <^
y
^
-
/
.■■/'
^^
-
/
/ ^
" G =
—
/ .- /■
/
/ /
G =
1
/••'
*" .^^
/
G =
l+0.5i - - ■
'^
G =
2— ■
'■■/
G =
l+2i --
f
1
1
1
1 1 1 1
1^/1,1
Figure 7.8: Breakwater displacement profiie for different G values at s = 0.9, "piH = 0.5,
h/H = 0.25 and EI/p2gH^ = 0.02.
7.6. NUMERICAL RESULTS AND DISCUSSION
141
y/H
h/H = 0.1
h/H = 0.25
h/H = 0.5
h/H = 0.75
h/H = 0.9
2 3
Figure 7.9: Breakwater displacement profile for different h/H ratios at s = 0.9, piH = 0.5,
EI/p2gH^ = 0.02 and G = 1.
K
0.5
0.4
0.3
0.2
0.1
f
G = —
G = l
G = 1+0.51
G = 2 --
G=l+2i -
0.05
0.1
EI/pgH
2
.4
0.15
0.2
Figure 7.10: Force coefficient versus EI/p2gH'^ for different G values at pjH = 0.5,
h/H = 0.25, and s = 0.9.
142
CHAPTER 7. WAVE PAST POROUS PLATE BREAKWATER
K
0.5
0.4
0.3
0.2
0.1
f
h/H = 0.1 —
h/H = 0.25
h/H = 0.5 --
h/H = 0.75 -.-.
h/H = 0.9 _..._
0.05
0.1
EI/pgH
2
.4
0.15
0.2
Figure 7.11: Force coefficient versus EI/p2gH'^ for different h/H ratios at pjH = 0.5,
s = 0.9 and G = 1 + 0.5i.
Chapter 8
WAVE TRAPPING BY FLEXIBLE
POROUS BREAKWATERS
8.1 INTRODUCTION
After analyzing wave scattering by flexible porous structures in the previous chapters,
wave trapping by flexible porous partial breakwaters is considered in a two-layer fluid.
Wave trapping in a two-layer fluid is a complex phenomena as it includes the trapping
of surface- and internal-waves simultaneously. In the present chapter, the efficiency of a
flexible porous partial plate breakwaters in trapping surface- and internal-waves near the
end-wall of a semi-inflnite long channel in a two-layer fluid domain is investigated based
on the linearized-theory of water waves.
8.2 DEFINITION OF THE PHYSICAL PROBLEMS
In the present section, physical problems for wave trapping by flexible porous partial
breakwaters in a two-layer fluid are considered. The two cases, bottom-standing and
surface-piercing breakwaters, are illustrated in Figs. 8.1 and 8.2 respectively. In the two-
layer fluid, the upper fluid has a free surface (undisturbed free surface located at y = 0)
and the two fluids are separated by a common interface (undisturbed interface located
143
144 CHAPTER 8. WAVE TRAPPING BY FLEXIBLE POROUS BREAKWATERS
Porous and
flexible breakwater
Free surface
Incident wave in
surface mode
X
y = H-b
y = h
Interface
Incident wave in
internal mode
y = H
Figure 8.1: Definition sketch for wave trapping by bottom-standing partial plate break-
water.
at y = h), each fiuid occupying the regions —L < x < +oo; < y < /i in case of the
upper fiuid of density pi and —L < x < +oo; h < y < H in case of the lower fiuid of
density p2- L^f and Lif represent the notations for the portion of the breakwater which
is in upper and lower fiuid domain respectively where as notations Lg and L^p represent
the regions, the gap and the portion of the breakwater which is above the free surface
respectively. The porous breakwater is located at a; = with L„/ = {H — b < y < h);
Lif = {h < y < H) for a bottom-standing breakwater and L^p = {—{hi — H) < y < 0);
Luf = {^ < y < h) and Lif = {h < y < H — b) for a surface-piercing breakwater. On the
other hand, the end-wall is located aX x = —L; < y < H (see Figs. 8.1 and 8.2).
8.3 MODEL FOR FLUID FLOW
The fiuid velocity potential will satisfy the condition (3.15) on the impermeable end- wall.
Hence
7— = on x = -L, < y < H.
ox
(8.1)
Considering the waves incident from large positive x upon the fiexible porous partial
breakwaters, the velocity potentials are obtained by eigenfunction-expansion method sim-
.3. MODEL FOR FLUID FLOW
145
Porous and
flexible breakwater
y = -(hi -H)
Incident wave in
surface mode
X
Free surface
Interface
2)
Incident wave in
internal mode y = h
y = H-b
y = H
Figure 8.2: Definition sketch for wave trapping by surface-piercing partial plate breakwa-
ter.
ilar to the case of a surface-piercing dike, in each of the two regions 1 and 2 as marked
in Figs. 8.1 and 8.2. The present fiuid problem is also a boundary value problem which
satisfies the fiuid fiow governing equation (3.1) along with the conditions Eqs. (3.12 —
3.14), (3.16 — 3.17) and (8.1). Applying the eigenfunction-expansion method, the velocity
potentials in the regions 1 and 2 are obtained and are given by
and
oo
h= J2 ^™ ^^•^ P^i^ + ^) fn{Pn, y), for -L <x <0,
n=IJI,l
II , oo .
^2 = J2 Ine'^"'''' fniPn, V) + J2 ^^ (i^"""" fn{Pn. V) , for X>0,
n=I n=I,II,l
(8.2)
(8.3)
The eigenfunctions /„'s are again same as defined in Eq. 6.3 and An, Rn {n = I, II, 1, 2, ...
are unknown constants to be determined.
In this case also only open water regions exist like in the previous two cases. Thus,
the wave numbers Pn (n = I, II for positive real roots and n = 1, 2, 3, ... for positive
purely imaginary roots) are the roots of the dispersion relation in p as given in Eq. 4.8.
The orthogonality condition as defined in the inner product Eq. 6.4 for the case of
membrane problem is also applicable in the present problem. The refiection coefficients in
146 CHAPTER 8. WAVE TRAPPING BY FLEXIBLE POROUS BREAKWATERS
SM and IM are defined in tlie case of membrane problem in Eq. 6.5 is also applicable in
the present problem. It may be noted that in the present case there are no transmission
coefficients because of the presence of the impermeable end-wall.
8.4 MODEL FOR FLEXIBLE PLATE RESPONSE
As described earlier in the general mathematical formulation, it is assumed that plate
breakwater is deflected horizontally with displacement C{y,t) = Re[({y)e~^'^*], where ^{y)
represents the complex deflection amplitude and is assumed to be small as compared to
the water depth. It is assumed that the plate breakwater is thin and behaves like a one-
dimensional beam of uniform flexural rigidity EI and mass per unit length nig. With
these assumptions, the governing equation (Eq. 3.20) relating the plate displacement ^
from equilibrium to that of differential pressure acting on the plate at a; = can be
applied. This gives
V '''^^
0, on y G Lop,
i^Pi(0i - 02)
■- P 1; = < ■
i^P2(
EI
^j , on y G L„j, (8.4)
on y G Lij,
where /3 is the structural frequency parameter as deflned by /3 = {mgUj'^ / Eiy/'^. The
breakwater will behave like a cantilever as it is assumed in the present study that the
bottom-standing breakwater is flxed at the seabed and is having a free edge inside the
fluid domain, and on the other hand, the surface-piercing breakwater is clamped above
the free surface and the free end is immersed inside the fluid domain.
The bottom-standing plate breakwater (Fig. 8.1) is flxed at the seabed and is having
a free edge inside the fluid domain. The corresponding boundary conditions as given in
Eqs. 3.22 and 3.23 can be applied. This gives
an) = 0, i'{H) = 0, i"{H -b) = and ({'")(// - b) = 0. (8.5)
On the other hand, the surface-piercing breakwater (Fig. 8.2) is clamped above the free
surface and the free end is immersed inside the fluid domain. The corresponding boundary
8.4. MODEL FOR FLEXIBLE PLATE RESPONSE 147
conditions as given in Eqs. 3.22 and 3.23 can be applied. This gives
a-{hi - H)) = 0, ei-ihi - H)) = 0, e'iH -b)=0 and {OiH - b) = 0. (8.6)
The continuity condition across the interface (Eq. 3.29) should be imposed. Hence
the deflection, slope of deflection, bending moment and the shear force acting on the plate
breakwater are continuous at the interface (the point on the breakwater where the two
fluids meet each other {x = 0; y = h)). This yield
ah-) = ah^), eih-) = eih^), eih-) = eih+), rih-) = rih^). (8.7)
In case of surface-piercing breakwater continuity condition across the free surface (Eq.
3.29) should be imposed. Hence, similar to the case of interface, the deflection, slope
of deflection, bending moment and the shear force acting on the plate breakwater are
continuous at the free surface (the point on the breakwater where the air and the upper
fluid meet each other {x = 0; y = 0)). This yield
e(0-)=aO+), {'(0-) = {'(0+), C"(0-) = nO+), r(0-) = r(0+). (8.8)
Applying the porous boundary condition (Eq. 3.36) on the porous plate breakwater
we obtain
= ikoG{(t)2 - 0i) + ioj^ {j = 1, 2) onx = 0,0 <y < H. (8.9)
dx
and G is the complex porous-effect parameter as defined in Eq. 3.37.
The condition across the gap Eq. 3.19 must be imposed. This gives
)i = 02, and — — = — — on X = 0, y G L„. (8.10)
ox ox
The conditions Eq. 8.9 and Eq. 8.10 can be rewritten as
on a; = 0, y G Lg,
koG{(j)2-(t>i) = { rl^. (8.11)
dx
■^ ia;{ for j = 1, 2 on a; = 0, y G L„j + Lif.
148 CHAPTER 8. WAVE TRAPPING BY FLEXIBLE POROUS BREAKWATERS
8.5 GENERAL SOLUTION PROCEDURE
Applying the continuity of (^^ (Eqs. 8.9 and 8.10) along the porous breakwater for y G
Luf + Lif on a; = and the gap y G L^ on x = , and invoking the orthogonality
relation (Eq. 6.4) over {0 < y < h) Li {h < y < H), we obtain
In- Rn = -AnSin PnL {u = I , II), R^ = - A^sinh p^L (n = 1, 2, ...). (8.12)
From Eq. 8.12 it is clear that the reflection coefficient Kr^ = — - = 1 when L = jXn/'2
^n
{n = I, II and j = 0, 1, 2, ...). It suggests that when the distance between the end-
wall and the breakwater is an integer multiple of half wavelength of the incident wave
in a particular wave mode (surface- or internal- wave mode), maximum reflection takes
place in the corresponding wave mode irrespective of breakwater conflgurations (bottom-
standing/surface-piercing, flexible/rigid, impermeable/permeable breakwater).
Utilizing the Eqs. 8.2, 8.3 and 8.12 a general solution is obtained for the fourth order
non-homogeneous ODE, Eq. 8.4 (partial plate breakwater governing equation) and is
given by
ay) = E.e'^y + E^e-'^y + E^e^^ + E,e-^y + fl HJofniPn, y)+
n=I
oo
J2 HnRnfn{Pn,y), (8.13)
n=I,II,l
where bar denotes the complex conjugate, the arbitrary constants Ei, i = 1, 2, 3, 4 and
II„ are deflned as below.
Ei = <
Eopi for y G Lop,
Euf.hiyeL^j, (8.14)
Elfi for y e Lif,
and
iujpAl — icot p„L)
Hn = 0, ye Lo, and H^ = Z\, , ^^, 0<y<H, (8.15)
and
15. GENERAL SOLUTION PROCEDURE
149
for y G L„p,
Pj = Wi for y G L„j, (8.16)
p2 for y e Lif.
Eopi, Eufi, Elf i for i = 1, 2, 3, 4 are unknown constants to be determined. Substituting
01, 02 and ( from Eqs. 8.2, 8.3 and 8.13 in Eq. 8.11 and utilizing relation (8.12), we
obtain
9oiy) + Yl 9niy)Rn = hrO<y<H,
n=I,IIA
where
r 11
9o{y) =
n=I
Py_
II
Y^ Inr-niy) + iujElfi e
n=I
iujEufi e '^^ for < y < /i.
lUJ
Elfi e-^y hi h<y< H,
(8.17)
(8.18)
9n{y) =
{GP^ + UB„)U{p^, y) for < y < h,
n = (/, //, 1, 2,
{GP^ + LB„)U{p^, y) for /i< y < H,
(8.19)
rn{y)
= <
{GPn + ZUB^)U{p^, y) for < y < h,
n = (/, //)
(GP„ + ZLB^)U{p^, y)ioTh<y< H.
(8.20)
The definition of the aforementioned notations and the range of their validity are as given
below
GP„ = 1 + icot p„L for y G Lg, (8.21)
UB^ = IGkoGP^ + ipr.
LBn = iGkoGPn + ipr>
u'^PiGP^
EI{pi - (3^)
UJ^p2GPn
for y ^ Luf,
for y e Lif,
(8.22)
(8.23)
150 CHAPTER 8. WAVE TRAPPING BY FLEXIBLE POROUS BREAKWATERS
ZUB^ = iGkoGPn - iPn - -rrrri — ^ f°^ y^L^f, (8.24)
and
ZLB^ = IGkoGP^ - ipn - Z^^f^l^ f°^ V ^ Lif. (8.25)
We can apply the least-squares-approximation method to Eq. 8.17 as described earlier
in general mathematical formulation chapter. We can write
N
Q{y) = 9o{y)+ E RnQniy), for O < y < //. (8.26)
n=I,II,l
Applying the least-squares method, we obtain
(•H _ dO(v)
/ Qiy)^W^dy = 0, hTn = I, II, 1, 2,...,iV, (8.27)
Jo OKn
where the bar denotes the complex conjugate.
Eq. 8.27 provides A^ + 2 linear equations with A + 10 unknowns in case of bottom-
standing breakwater and A + 14 unknowns in case of surface-piercing breakwater, as ho{y)
involves extra unknowns Eopi, Eufi, Elfi for i = 1, 2, 3, 4. In case of bottom-standing
breakwater 8 more equations are needed to solve the matrix systems and the required
another 8 linear equations are obtained by substituting the expression for { from Eq. 8.13
in the breakwater end conditions Eq. 8.5 and interface continuity conditions Eq. 8.7.
On the other hand, in case of surface-piercing breakwater the required 12 more equations
are obtained by substituting the expression for { from Eq. 8.13 in the breakwater end
conditions Eq. 8.6, free surface and interface continuity conditions Eqs. 8.8 and 8.7.
These system of equations are solved using Gauss-elimination method to compute and
analyze various physical quantities of interest. Number of evanescent modes in the series
are selected based on the experience of the numerical convergence experiment.
8.6 NUMERICAL RESULTS AND DISCUSSION
In general, the behavior of reflected wave energy is one of the major criteria in deciding
the effectiveness of a breakwater in trapping the water waves. It is well known in the
literature that trapping occurs between a breakwater and the end-wall for specific wave
8.6. NUMERICAL RESULTS AND DISCUSSION 151
frequencies, which leads to minimum wave reflection by the breakwater. In the present
study, numerical results are computed to investigate the performance of both bottom-
standing and surface-piercing partial breakwaters in trapping the surface- and internal-
waves in a two-layer fluid by keeping the mass per unit length nig of the breakwater
flxed at 10 Kg/m^. The influence of various physical parameters like water depth h/H,
fluid density ratio s, porous-effect parameter G of the breakwater, length of submergence
of breakwater {H — b)/H, total length of breakwater hi/ H in case of surface-piercing
breakwater, breakwater flexural rigidity EI on wave trapping in both SM and IM wave
motion, and the nature of hydro dynamic force Fr and breakwater deffection { are studied.
The deflnition of hydrodynamic force Fr is given by
Fr = ioo f pAMo,y) - Mo,y)]dy. (8.28)
It is observed that in both the cases of bottom-standing and surface-piercing break-
waters reflection coefficients in SM, Krj and in IM, Kr^ are periodic in nature for
normalized breakwater positions L/A/ and L/\n respectively. Full reflection takes place
at L/\j = n/2, for n = 1, 2, ..., j = I, II and minimum reflection occurs at an in-
termediate points of (n — l)/2 < L/Xj < n/2, for n = 1, 2, ... and {j = I, II). A
similar phenomenon of minimum reflection in case of single-layer fluid is referred as wave
trapping in the literature (see Sahoo et al. (2000) and Yip et al. (2002)).
8.6.1 The Case of a Bottom-Standing Breakwater
The influence of various important physical parameters on trapping of both surface- and
internal-waves by bottom-standing partial breakwaters is discussed in this subsection.
Wave Reflection
Fig. 8.3 (a) and (b) depict the effect of bottom-standing breakwater stiffness EI on
reffection coefficients Krj in SM and Krn in IM respectively. In Fig. 8.3 (a), minimum
wave reffection and hence high wave trapping is observed in the case of waves in SM for
higher values of EI. Since smaller values of EI enhances wave transmission across the
152 CHAPTER 8. WAVE TRAPPING BY FLEXIBLE POROUS BREAKWATERS
breakwater, wave trapping is observed to be negligible. However, the difference in the
reflection coeflicients for large EI values (EI = 40, 100 Nm) is marginal. It is due to the
fact that after certain rise in the value of flexural rigidity, the breakwater starts behaving
like a rigid wall and the reflection coefficient is not affected with further rise in breakwater
rigidity. On the other hand, in the case of wave motion in IM the inffuence of change in
breakwater flexural rigidity is insigniflcant (Fig. 8.3 (b)).
The variation of reflection coefficients versus L/ Xj for (j = I, II) in both the cases
of SM and IM are plotted in Fig. 8.4 (a) and (b) respectively for different values of the
porous-effect parameter G. In Fig. 8.4 (a), the amount of trapped waves in SM is found
to be increasing with a decrease in the porous-effect parameter |G|. On the other hand,
in the case of wave motion in IM, the amount of trapped waves is found to be less for
both small and large value of |G| (Fig. 8.4 (b)). Highest wave trapping in IM is observed
for moderate values of porous-effect parameters G = 1 + li and G = 2. Interestingly,
for both SM and IM, wave motion at G = 1 + li and G = 2, the minimum value of
the reffection coefficient is same in magnitude and the only difference is the breakwater
location at which the minimum reffection coefficient is observed. This may be due to the
change in phase angle of the porous-effect parameter.
The inffuence of non-dimensional breakwater length h/ H of the bottom-standing break-
water on the reffection coefficients in SM and IM are shown in Fig. 8.5 (a) and (b) re-
spectively. In Fig. 8.5 (a) for SM wave motion, highest wave trapping is observed when
the breakwater length, h/ H is just exceeds the depth of the lower ffuid, (1 — h/ H) (at
h/ H = 0.6). On the other hand, the change in breakwater length has negligible effect to
trap the waves in IM in the region between the breakwater and the channel end-wall (Fig.
8.5 (b)).
The inffuence of interface location h/H on the reffection coefficients in SM and IM
are shown in Fig. 8.6 (a) and (b) respectively. Highest wave trapping is observed in SM
wave motion when interface is nearer to the seabed. However, for h/H = 0.25 and 0.5
the reffection coefficients are found to be same (Fig. 8.6 (a)). On the other hand, highest
wave trapping is observed in IM wave motion when the interface is located just at the
8.6. NUMERICAL RESULTS AND DISCUSSION 153
center of free surface and the seabed (Fig. 8.6 (b)).
The reflection coefficients versus L/ Xj for {j = I, II) are plotted in SM and IM for
various values of s in Fig. 8.7 (a) and (b) respectively. In Fig. 8.7 (a) wave trapping in
SM is found to be increasing with the increase in the value of s. On the other hand, the
reflection coefficient in IM follows an opposite trend compared to the case of SM wave
motion and the wave trapping is found to be decreasing with the increase in the value of
s (Fig. 8.7 (b)).
Breakwater Deflection and Hydrodynamic Force
The bottom-standing breakwater deflection and hydrodynamic forces are plotted in Figs.
8.8 and 8.9. Except for h/ H = 0.8, in general the breakwater deflection decreases with
the decrease in the value ofb/H (Fig. 8.8 (a)) and in Fig. 8.8 (b), breakwater deflection
increases with the decrease in the value of h/H. In Fig. 8.9 (a), the hydrodynamic force
is decreasing with the increase in the value of G. However, higher hydrodynamic forces
are observed for G with higher values of inertial effect of ffuid inside the porous media,
which is represented by the imaginary part of G. On the other hand, the hydrodynamic
force is increasing with the increase in the value of s (Fig. 8.9 (b)), which is due to the
high inertial wave amplitude as s approaches to one (see Kundu and Cohen (2002)).
8.6.2 The Case of a Surface-Piercing Breakwater
The influence of various important physical parameters on trapping of both surface- and
internal-waves by surface-piercing partial breakwaters is discussed in this subsection.
Wave Reflection
Fig. 8.10 (a) and (b) depict the effect of surface-piercing breakwater stiffness EI on
reffection coefficients Krj in SM and Krjj in IM respectively. In Fig. 8.10 (a) minimum
wave reffection and hence high wave trapping is observed for moderate value of breakwater
ffexural rigidity {EI = 40 Nm). On the other hand, in the case of wave motion in IM
154 CHAPTER 8. WAVE TRAPPING BY FLEXIBLE POROUS BREAKWATERS
the maximum wave trapping is observed for high value of breakwater flexural rigidity
{EI = 100 Nm) (Fig. 8.10 (b)).
The variation of reflection coefficients versus normalized breakwater position L/Xj
for {j = I, II) in both the cases of SM and IM are plotted in Fig. 8.11 (a) and (b)
respectively for different values of the porous-effect parameter G. In Fig. 8.11 (a), the
amount of trapped waves in SM is found to be increasing with a decrease in the porous-
effect parameter |G|. On the other hand, in the case of wave motion in IM the amount of
trapped waves is found to be less for both small and large value of |G| (Fig. 8.11 (b)).
The influence of non-dimentional length of submergence 1 — h/ H in the case of surface-
piercing breakwater on wave trapping in both SM and IM are shown in Fig. 8.12 (a) and
(b) respectively. In SM wave motion, the amount of wave energy trapped is found to be
increasing with an increase in 1 — h/ H . However, the numerical value of minimum of the
reflection coefficients for l — b/H = 0.6 and 0.7 are same. This is because most of the wave
energy in SM concentrated in the upper fluid and this wave energy decays exponentially
towards the seabed (Fig. 8.12 (a)). On the other hand, wave trapping in IM is found to
be high when the length of submergence of the breakwater is equal or grater than the
depth of upper fluid (Fig. 8.12 (b)).
The effect of change in non-dimensional length hi/ H of the surface-piercing breakwater
on the reflection coefficients in SM and IM are shown in Fig. 8.13 (a) and (b) respectively.
In SM wave motion, the amount of wave energy trapped is found to be increasing with
a decrease in the value of hi/ H (Fig. 8.13 (a)). On the other hand, in case of IM wave
motion an opposite trend is observed, where the wave trapping is found to be increasing
with an increase in the value of hi/ H (Fig. 8.13 (b)).
The influence of depth ratio h/ H on the reflection coefficients in SM and IM are
shown in Fig. 8.14 (a) and (b) respectively. Highest wave trapping in SM wave motion is
observed when interface is located in between the free surface and breakwater free edge
(Fig. 8.14 (a)). On the other hand, similar to the case of bottom-standing breakwater,
highest wave trapping is observed in IM wave motion when the interface is located just
at the center of free surface and the seabed (Fig. 8.14 (b)).
8.6. NUMERICAL RESULTS AND DISCUSSION 155
The reflection coefiicients versus the normahzed breakwater position L/Xj for {j =
I, II) are plotted in SM and IM for various values of s in Fig. 8.15 (a) and (b) respectively.
In Fig. 8.15 (a), wave trapping in SM is found to be higher for smaller values of s.
However, it is observed that the value of wave reflection in SM is same for s = 0.5 and
0.25. On the other hand, the reflection coefficient in IM is found to be high for the
intermediate values of s (= 0.5) (Fig. 8.15 (b)).
Breakwater Deflection and Hydrodynamic Force
The surface-piercing breakwater deflection and hydrodynamic forces are plotted in Figs.
8.16 and 8.17. The breakwater deflection is increasing with the increase in the value of
G (Fig. 8.16 (a)) and hi/H (Fig. 8.16 (b)). In Fig. 8.17 (a) the hydrodynamic force
is increasing with the increase in the value of hi/H. However, the hydrodynamic forces
attain a constant value for all hi/ H values at higher value of breakwater flexural rigidity.
It is expected as at higher values of breakwater rigidity the breakwater starts behaving
like a rigid wall. On the other hand, the hydrodynamic force is high for small and large
values of s (Fig. 8.17 (b)).
8.6.3 Summary of Important Observations
In both the cases of bottom-standing and surface-piercing breakwaters, reflection coeffi-
cients in SM and IM are periodic in nature for normalized breakwater positions. Full reflec-
tion takes place at L/\j = n/2, for n = 1, 2, ..., j = I, II and minimum reflection occurs
at an intermediate points of (n — l)/2 < L/ Xj < n/2, for n = 1, 2, ... and {j = I, II).
A similar observation is reported in single-layer fluid (see Sahoo et al. (2000) and Yip
et al. (2002)). The important observations from the present numerical results for wave
trapping by porous partial breakwaters are summarized pointwise as below:
The Case of a Bottom-Standing Breakwater
1. High wave trapping for waves in SM is observed for higher values of EI and the
influence of EI on wave motion in IM is insigniflcant.
156 CHAPTER 8. WAVE TRAPPING BY FLEXIBLE POROUS BREAKWATERS
2. The amount of trapped waves in SM is increasing with a decrease in the porous-
effect parameter and highest wave trapping in IM is observed for moderate values
of porous-effect parameters.
3. For G = 1 + li and G = 2, the minimum values of the reflection coefficient in SM
and IM are same in magnitude and the only difference is the breakwater location
at which the minimum reflection coefficient is observed. This may be due to the
change in phase angle of the porous-effect parameter.
4. Highest wave trapping in SM is observed when the breakwater length, b/H just
exceeds the depth of the lower ffuid. The change in breakwater length has negligible
effect on trapping of waves in IM in the region between the breakwater and the
channel end-wall.
5. Highest wave trapping is observed in SM wave motion when interface is nearer to
the seabed and highest wave trapping is observed in IM wave motion when the
interface is located just at the center of free surface and the seabed.
6. Wave trapping in SM is found to be increasing with the increase in the value of s
and the wave trapping in IM is found to be decreasing with the increase in the value
of s.
7. Except for b/H = 0.8, in general the breakwater deffection is decreasing with the
decrease in the value of b/H .
8. Breakwater deffection is increasing with the decrease in the value of h/H.
9. Hydrodynamic force is decreasing with the increase in the value of G. However,
higher hydrodynamic forces are observed for G with higher values of inertial effect
of ffuid inside the porous media, which is represented by the imaginary part of G.
10. Hydrodynamic force is increasing with the increase in the value of s.
The Case of a Surface-Piercing Breakwater
1. High wave trapping in SM is observed for moderate value of breakwater ffexural
rigidity and in the case of wave motion in IM the maximum wave trapping is observed
for high value of breakwater ffexural rigidity.
.6. NUMERICAL RESULTS AND DISCUSSION 157
2. The amount of trapped waves in SM is increasing with a decrease in the porous-
effect parameter and in the case of wave motion in IM the amount of trapped waves
is less for both small and large values of \G\.
3. The amount of wave energy trapped in SM is found to be increasing with an in-
crease in 1 — b/H and wave trapping in IM is found to be high when the length of
submergence of the breakwater is equal or greater than the depth of the upper fluid.
4. The amount of wave energy trapped in SM is found to be increasing with a decrease
in the value of hi/ H and the wave trapping in IM is found to be increasing with an
increase in the value of hi/H.
5. Highest wave trapping in SM wave motion is observed when interface is located in
between the free surface and breakwater free edge. Similar to the case of a bottom-
standing breakwater, highest wave trapping is observed in IM wave motion when
the interface is located just at the center of free surface and the seabed.
6. Wave trapping in SM is found to be higher for smaller values of s and the reflection
coefficient in IM is found to high for the intermediate value of s.
7. Breakwater deflection is increasing with the increase in the value of G and hi/H.
8. Hydro dynamic force is increasing with the increase in the value of hi/H .
9. Hydro dynamic force is high for small and large values of s.
158 CHAPTER 8. WAVE TRAPPING BY FLEXIBLE POROUS BREAKWATERS
Kr,
0.2
EI = 10 Nm
EI = 20 Nm
EI = 40 Nm
EUlOONm
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
L/h
0.4
0.2
EUlONm
EI = 20 Nm
(a)
0.5
EI = 40Nm --
EUlOONm- -
1
1.5
II
(b)
Figure 8.3: Reflection coeflicients in (a) SM, Krj versus L/A„ and (b) IM, Krn versus
L/ Xjj for bottom-standing breakwater at different EI values, b/H = 1.0, G = 2, h/H =
0.25, and s = 0.75.
.6. NUMERICAL RESULTS AND DISCUSSION
159
G = l+0.5i G = 2
J I I I I I I L
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
L/X,
0.6
Kr
II
0.4
0.2
(a)
' t)cf^! ^^V
I 1 A
V ■i..''
G = \ —
G = 1+0.51
G = l+li --
G = 2
G = 10
0.5
1
1.5
II
(b)
Figure 8.4: Reflection coeflicients in (a) SM, Kri versus L/A„ and (b) IM, Kru versus
L/Xjj for bottom-standing breakwater at different G values, h/H = 0.25, b/H = 1.0,
EI = 10 Nm, and s = 0.75.
160 CHAPTER 8. WAVE TRAPPING BY FLEXIBLE POROUS BREAKWATERS
0.4
0.2
\f b/H=l ■. -b/H^O.Qi /b/H = 0.8\.
b/H = 0.7 b/H = 0.6 b/H = 0.5
I I I
0.5
1
1.5
0.8
0.6
Kr
II
0.4
0.2
(a)
\
A A
h i
:y
%>-':■<</•• V"^ ■
1
y
- b/H=l
b/H = 0.8 --
b/H = 0.6 -- "
b/H = 0.5
b/H = 0.7 — ■
1 1
b/H = 0.5 -■-
1
0.5
1
L/X
1.5
II
(b)
Figure 8.5: Reflection coeflicients in (a) SM, Krj versus L/X^ and (b) IM, Krn versus
L/ Xjj for bottom-standing breakwater at different b/H values, h/H = 0.5, G = 2, EI =
10 Nm, and s = 0.75.
.6. NUMERICAL RESULTS AND DISCUSSION
161
0.2
h/H = 0.25
0.5
h/H = 0.5 - - h/H = 0.75
1
1.5
(a)
h/H = 0.25 — h/H = 0.5 - - ■ h/H = 0.75
0.5
1
L/X
1.5
II
(b)
Figure 8.6: Reflection coefficients in (a) SM, Krj versus L/A„ and (b) IM, Krn versus
L/ Xjj for bottom-standing breakwater at different h/H values, b/H = 1.0, G = 2, EI =
10 Nm, and s = 0.75.
162 CHAPTER 8. WAVE TRAPPING BY FLEXIBLE POROUS BREAKWATERS
(a)
KriiO.5
(b)
Figure 8.7: Reflection coeflicients in (a) SM, Krj versus L/A„ and (b) IM, Krn versus
L/ Xjj for bottom-standing breakwater at different s values, h/H = 0.25, G = 2, EI =
10 Nm, and b/H = 1.0.
.6. NUMERICAL RESULTS AND DISCUSSION
163
y/H
\J
1 1 1 1 1 1 1 1
1 . b/H = 1.0 —
0.2
- / / b/H = 0.9
/ / b/H = 0.8 --■ /'
0.4
••••■" ■"■-• -- '
/ .• ^. ..' /
..^-■•"^•s
/ ; ..-••' \ .-^ ^
0.6
^■^ ^■■■■' - - "
■■' .■-• ^c'-'-; ' '
C-^-^ ^..,^..--"";.-' b/H = 0.7 — ■
0.8
- y/ ^^^^]f:r^" b/H = 0.6 -- -
/• ..-r:^- v - ' b/H = 0.5 - -
■^^- ^
1
>- 1 1 1 1 1 1 1 1
0.01
0.02
0.03
0.04
(a)
y/H
0.2
0.4
0.6
0.8
1
1 1 '1
/
/
/
/
/
/
/ ^
1 1 V ^ y
' x'''^^
h/H = 0.25 —
/y^
h/H = 0.5
/
h/H = 0.75 --■
(
1 1 1 1 1
0.001
0.002
0.003 0.004
(b)
Figure 8.8: Barrier deflection, ^ at (a) hjH = 0.25 and (b) b/H = 1.0 for bottom-standing
breakwater with s = 0.75, G = 1, EI = 20 Nm, and L/Xj = 0.25.
164 CHAPTER 8. WAVE TRAPPING BY FLEXIBLE POROUS BREAKWATERS
60
Fr/Io 30
G=l+0.5i
G=l+li-
G = 2-
G=10-
100
EI (Nm)
150
200
40
30
Fr/L 20
10
(a)
/
/
1 1
/^ —
-
//
" /'
s = 0.25 —
- /' ,.■■'
s = 0.5
- /
s = 0.75 --■
f
1 1
50
100
EI (Nm)
150
200
(b)
Figure 8.9: Hydro dynamic force, Fr at (a) s = 0.75 and (b) G = 2 for bottom-standing
breakwater with h/H = 0.25, b/H = 1.0, and L/A/ = 0.25.
.6. NUMERICAL RESULTS AND DISCUSSION
165
Kr,
0.8
0.6
0.4
0.2
/\
/\ /
i f
\ r \
Ij \ j-
// Vv
/■' \A /'
\ r
(■ /■' \
\ M \\ /'
\ i?
\\ // \
* //:' \i h
V /•'
% I ^
Ah U /■'
\ r
\: .••'/ •■'
u\ /■ /■' \\\/i }
V ■•'/■''
V ■r'
xj 1
fcj''
^■■■'l s^-/ _
S-
\/
v^ V
EUlONm
— EI =
^ 40 Nm - - ■
" EI = 20Nm
1
EI =
1
^lOONm --
0.5
1
1.5
0.4
0.2
(a)
EUlONm
EI = 20 Nm
EI = 40 Nm - - ■
EUlOONm --
0.5
1
1.5
11
(b)
Figure 8.10: Reflection coefficients in (a) SM, Krj versus L/Xn and (b) IM, Kru versus
L/Xjj for surface-piercing barrier at different EI values, b/H = 0.5, hi/H = 1.0, G = 2,
h/H = 0.5, and s = 0.75.
166 CHAPTER 8. WAVE TRAPPING BY FLEXIBLE POROUS BREAKWATERS
Kr
Kr,
0.4
0.2
(a)
••V, / '■...•■; / i-.// /
.1 I
^/y^y' I A_ ,/■ wV_/
K /■
/ ■-■
\/
G=l
G = l+li --■ G=10--
G=l+0.5i G = 2
I L
0.5
1
1.5
II
(b)
Figure 8.11: Reflection coefficients in (a) SM, Krj versus L/A„ and (b) IM, Krji versus
L/ Xjj for surface-piercing barrier at different G values, h/H = 0.5, b/H = 0.7, hi/ H =
1.0, EI = 10 Nm, and s = 0.75.
.6. NUMERICAL RESULTS AND DISCUSSION
167
Kr
1
0.8
0.6
0.4
0.2
\ r
\ /9\ /7
\\ i \\ f
\ u
V- I- V- ■
- \ ft
\ >
\- /^ I- / -
\ i
\\ /•' \\ /■'
\ '
\ H
\ /l' % /^
1 /'
\ -1
V- / •'' v'- / ■ ~
\ m
fi /;.
V'- 1 -^ \\' / ■'
\ Pl
i 1
V- / ■ \\ / ^
\//
\ u
S-^ / \: 1
- y,
V
\ 1 \ i -
_ (H-b)/H --
= 0.5 —
(H-b)/H = 0.7 - - ■ _
(H-b)/H --
-- 0.6
1
1
0.5
1
1.5
0.4
0.2
(a)
(H-b)/H = 0.5
(H-b/H = 0.6
(H-b)/H = 0.7--
0.5
1
1.5
II
(b)
Figure 8.12: Reflection coefficients in (a) SM, Krj versus L/A„ and (b) IM, Krji versus
L/Xii for surface-piercing barrier at different h/ H values, h/ H = 0.5, G = 2, hi/ H = 1.0,
EI = 10 Nm, and s = 0.75.
168 CHAPTER 8. WAVE TRAPPING BY FLEXIBLE POROUS BREAKWATERS
Kr,
0.8
0.6
0.4
0.2
A /
'\ A /
V.
\ /
\ \ /
y
V I/:
h^/E =
1.0 —
hj/H=1.5 --■
hj/H =
1.25
h/H=1.75---
1 1
0.5
1
1.5
(a)
h,/H=1.5 --
h,/H=1.75--
(b)
Figure 8.13: Reflection coefficients in (a) SM, Krj versus L/X^ and (b) IM, Kru versus
L/ Xjj for surface-piercing barrier at different hi/ H values, h/ H = 0.5, G = 2, b/H = 0.5,
EI = 10 Nm, and s = 0.75.
.6. NUMERICAL RESULTS AND DISCUSSION
169
Kr
1
0.8
0.6
0.4
0.2
\
A
.
A
li
1
\
\
J
\ '/
M
1
\ 1
- I
1
\ '/
A
il
n ' "
V
f
I '/
A
/I
i\ '/
X
1
\j
V
1 /•
/ /
/ /
/ /
/ /•
V:
h/H =
= 0.1-
— h/U =
0.3 ■
1
h/H
= 0.5 - - ■ _
1
0.5
1
hll
1.5
(a)
h/H = 0.1 — h/H = 0.3 h/H = 0.5 - -
0.5
1
L/X
1.5
II
(b)
Figure 8.14: Reflection coefficients in (a) SM, Kri versus L/A„ and (b) IM, Krn versus
L/Xii for surface-piercing barrier at different h/H values, h/H = 0.5, hi/ H = 1.0, G = 2,
EI = 10 Nm, and s = 0.75.
170 CHAPTER 8. WAVE TRAPPING BY FLEXIBLE POROUS BREAKWATERS
Kr
1
0.8
0.6
0.4
0.2
/
\/
\ /
\ 1
1- /
■ V
s = 0.25 -
s = 0.5 ■
s = 0.75 -
0.5 1
L/L
1.5
(a)
(b)
Figure 8.15: Reflection coefficients in (a) SM, Kri versus L/\n and (b) IM, Krn versus
L/\ii for surface-piercing barrier at different s values, h/ H = 0.25, G = 2, EI = 10 Nm,
hi/H = 1.0 and b/H = 0.5.
.6. NUMERICAL RESULTS AND DISCUSSION
171
y/H
0.2
0.4
0.6
0.8
1
k '*x.
1 1 1
1 1 1
\ ^■
Y* ***■
\x
G =
1 —
•.^_
G =
l+0.5i
X yy
G =
1+li --■
,y
-
G =
2 — •
-
1 1
G =
10 --
1 1 1
1 1 1
0.2
0.4 0.6
0.8
-1
-0.5
y/H
0.5
(a)
h /H=1.0 —
hj/H=1.25
h/H=1.5 --■
hj/H=1.75--
0.5 1
1.5 2 2.5 3 3.5
(b)
Figure 8.16: Barrier deflection, { at (a) hi/H = 1.0 and (b) G = 2 for surface-piercing
breakwater with h/H = 0.5, b/H = 0.7, EI = 10 Nm, s = 0.75 and L/A/ = 0.25.
172 CHAPTER 8. WAVE TRAPPING BY FLEXIBLE POROUS BREAKWATERS
Fr/In
30
/'/'.••■
hj/H^l.O —
hi/H=1.25
hj/H=1.5 --
h,/H=1.75--
50
100
EI(Nm)
150
200
(a)
Fr/L
40
30
20
10
1 1 1
f
-
/
- /
-
,' / ■
;/
:;
r
s = 0.25 —
s = 0.5
s = 0.75---
1 1
50
100
EI (Nm)
150
200
(b)
Figure 8.17: Hydro dynamic force, Ft at (a) s = 0.75 and (b) hi/H = 1.0 for surface-
piercing breakwater with h/H = 0.25, b/H = 0.7, G = 2 and L/Xj = 0.25.
Chapter 9
CONCLUDING REMARKS
9.1 SALIENT FEATURES OF THE STUDY
The salient features of the present research work are summarized as follows:
1. Generalization of the Single-Layer Fluid Structure Interactive Models to
Two-Layer Systems
Models based on linearized-theory of water waves have been developed to generalize
a class of 2-D wave-structure interaction problems in a single-layer fluid to a two-
layer fluid. In a single-layer fluid a number of models have been developed over the
years to solve wave-structure interaction problems. In the case of a two-layer fluid,
time-harmonic waves can propagate with two different wave numbers, which makes
the mathematical modeling and analysis difficult. In the present study it is as-
sumed that the ffuid is inviscid and incompressible, and the effect of surface tension
is neglected. Special orthogonal/orthonormal relations suitable for the two-layer
fluid are introduced to simplify the equations in the mathematical models. Com-
puted results in the study are compared with the available results in the single-layer
fluid. Furthermore, the numerical results from two different methods (matched-
eigenfunction-expansion method and WSAM) are also compared with each other.
The results show a close agreement when the structures are widely placed.
173
174 CHAPTER 9. CONCLUDING REMARKS
2. Investigation of Surface- and Internal- Wave Scattering
An attempt has been made to understand the scattering of surface- and internal-
waves in a two-layer fluid. Scattering of water waves has been and continues to
be a subject of much research. From the practical side, it is essential for many
important ocean engineering problems to consider surface- and internal- waves to-
gether. From the research aspect, interest lies in understanding the complex flow
physics. In the present work scattering of surface- and internal-waves by a single
and a pair of identical rectangular dikes in a two-layer fluid is analyzed in two-
dimensions within the context of linearized-theory of water waves. Both the cases
of surface-piercing and bottom-standing dikes are considered. The scattering anal-
ysis for surface- and internal-waves has been extended for the wave past flexible
porous structures, namely (i) Flexible porous membrane and (ii) Flexible porous
plate. The numerical results are generated and are analyzed to understand the
effect of various physical parameters on scattering for surface- and internal- waves.
3. Evaluation of Efficiency of Flexible Porous Breakwaters
In connection with the design of a breakwater, the functional performance of and the
environmental load on the structure under various incident wave conditions must be
known a priori. A number of studies have been reported in the literature for evalu-
ation of efficiency of rigid breakwaters. Due to the rapid developments in material
science, flexible structures are believed to be extremely effective as breakwaters,
absorbing or reffecting much of the wave energy. In resonance, wave energy must be
radiated or dissipated so that the oscillation can be effectively suppressed. Recently
porous breakwaters are proposed to tackle such difficult practical problems. In the
present research work mathematical models are developed to investigate the effi-
ciency of ffexible porous breakwaters in a two-layer ffuid. Analysis of surface- and
internal-wave dissipation, structural response and hydrodynamic force on ffexible
porous breakwaters (porous membrane and plate) are considered for various physi-
cal parameters in a two-layer ffuid. The important observations from the numerical
9.1. SALIENT FEATURES OF THE STUDY 175
results are elaborated.
4. Study of Trapping Phenomena for Surface- and Internal- Waves
The study of wave trapping in various physical situations is important as it has
applications in many coastal engineering applications like dynamics and sedimen-
tology of the near shorezone through their interaction with ocean swells and surfs.
In recent years, there is a tremendous interest in using partial breakwaters to con-
trol waves. Many researches in past studied the wave trapping phenomenon due to
porous breakwaters at a finite distance from the end-wall. They observed that the
wave refiection depends very much on the distance between the barrier and the end-
wall. Moreover, to widen the range of wave trapping and reduce the wave force on
the breakwater, fiexibility is introduced on the breakwaters. In the present thesis,
the trapping of surface- and internal-waves by porous and fiexible partial breakwa-
ters near the end of a semi-infinitely long channel is studied in a two-layer fiuid
of finite depth having a free surface- and an interface. In the study both surface-
piercing and bottom-standing configurations are considered. The wave trapping for
both surface- and internal-waves is analyzed for various physical parameters.
5. The Identification and Evaluation of Fluid Density Ratio and Interface
Location as Two Major Physical Parameters Influencing Effectiveness of
Breakwaters
One of the major objectives of present study is to investigate the infiuence of the
fluid density ratio and interface location on the effectiveness of different breakwaters.
It is observed and identified in the present study that the fiuid density ratio and the
interface location plays a vital role in the performance of the various breakwaters
considered in the study. The infiuence of fiuid density ratio and interface location
on the different breakwater performance is described based on the observations from
the numerical results generated from this thesis.
176 CHAPTER 9. CONCLUDING REMARKS
9.2 FUTURE SCOPE OF RESEARCH
This section discusses the possible extensions of the present investigation. They are
presented below:
1. The immediate extension of the present research work is the three-dimensional and
oblique water wave scattering in a two-layer fluid.
2. Present analysis can be extended to study the effect of current in a two-layer fluid.
3. Investigations can be made on the surface- and internal- wave scattering and trapping
under a floating ice in a two-layer fluid.
4. The analysis of Very Large Floating Structures (VLFS) in a two-layer fluid is also
an area which may be exploited.
5. Investigation of surface tension effect of water waves in the two-layer fluid can be
an interesting extension of the present work.
6. Study of porous flexible wave maker problem in a two-layer fluid is an important
area because of its immediate practical applications.
7. Though some work has been reported on high wave energy dissipating flexible wave
chamber, no real investigation has been carried out in two-layer fluid wave mo-
tion. So there is a great scope to investigate the performance of high wave energy
dissipating flexible wave chamber, in a two-layer fluid.
8. Development of a numerical scheme to analyze the wave interaction with floating
and submerged structures of arbitrary geometry in a two-layer fluid using boundary
integral equation method will be an interesting extension of the present investigation
from the practical application point of view.
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LIST OF PUBLICATIONS FROM
THE PRESENT THESIS WORK
The thesis comprises of the following contributory papers:
Peer Reviewed
1. Suresh Kumar, P., and Sahoo, T., (2006). Wave interaction with a flexible
porous breakwater in a two-layer fluid. Journal of Engineering Mechanics,
ASCE, 132 (9), 1007-1014.
2. Suresh Kumar, P., Manam, S. R., and Sahoo, T., (2007). Wave scattering by
flexible porous vertical membrane barrier in a two-layer fluid. Journal of
Fluids and Structures, 23 (4), 633—647.
3. Suresh Kumar, P., Bhattacharjee, J., and Sahoo, T., (2007). Scattering of sur-
face and internal waves by rectangular dikes. Journal of Offshore Mechanics
and Arctic Engineering, ASME, 129 (4), 306-317.
Manuscript in Preparation
1. Suresh Kumar, P., and Sahoo, T., (2007). Trapping of surface and internal
waves by porous and flexible barriers in a two-layer fluid. Journal of Applied
Mechanics, ASME, (Communicated).
Conference Proceedings
1. Suresh Kumar, P., and Sahoo, T., (2005). Wave trapping by flexible porous
partial barriers in a two-layer fluid. Proceeding of 15th International Offshore
and Polar Engineering Conference and Exhibition, Seoul, Korea, 3, 572—579.
2. Suresh Kumar, P., and Sahoo, T., (2004). Wave past flexible porous break-
water in a two-layer fluid. Proceeding of 3rd Indian National Conference on
Harbor and Coastal Engineering, National Institute of Oceanography, Dona Paula,
Goa, India, 451-467.
3. Suresh Kumar, P., Bhattacharjee, J., and Sahoo, T., (2004). Wave interaction
with floating and submerged rectangular dikes in a two-layer fluid. Pro-
ceeding of International Conference on Conservation, Restoration, and Management
of Lakes and Coastal Wetland, Bhubaneswar, India, Paper No. OR06_02.
4. Suresh Kumar, P., Manam, S. R., and Sahoo. T., (2004). Wave scattering
by flexible breakwater in a two-layer fluid. Proceeding of National Workshop
on Advances in Fluid Dynamics and Applications, Utkal University, Bhubaneswar,
India.
ABOUT THE AUTHOR
f^ ^
Mr. P. Suresh Kumar, born on 23rd June, 1978 is a bachelor. He is Indian by Nationality.
Having passed the Higher Secondary (10+2) Examination of the Orissa Council of Higher
Secondary Education, Bhubaneswar, in the first Division in 1995, he took his degree of
B.E. in Mechanical Engineering with first class in the year 2000 from Utkal University,
Orissa. He obtained his masters degree (M.Tech) with first class from Mechanical Engi-
neering Department, Indian Institute of Technology, Guwahati with specialization in Fluid
and Thermal Science in 2003. After post graduation he entered into the Doctoral Study
in the Department of Ocean Engineering and Naval Architecture, at Indian Institute of
Technology, Kharagpur in the year 2003. The author has had excellent opportunity to
carry out research in a variety of areas in the fiuid and thermal science during his graduate
studies, including numerical modeling of fiow past porous and fiexible structures; wave
motion in a two— layer fiuid; scattering of water waves; internal waves; wave trapping;
fiuid structure interaction; fiow physics in large— hydraulic— diameter ducts; heat transfer
augmentation in heat exchangers; and, design and development of Savonius wind turbine
blades. He has to his credit some publication in the aforementioned areas of fiuid and
thermal science. He has considerable interest in yoga practice (Bhakti— Yoga), indoor
games and model making. Necessary particulars such as address for correspondence and
list of publications outside the present thesis work are given below.
ADDRESS FOR CORRESPONDENCE:
Present Address:
Invited Scientist
Coastal Engineering Research Department.
Korea Ocean Research &: Development Institute,
Ansan P.O.Box 29, Seoul - 425 600, Korea
E— mail: suresh_bbsr2000@yahoo. com,
jaga.suresh@gmail.com
Phone: (+82) 31-400-7818
Fax: (+82) 31-408-5823
Permanent Address:
C/0: Mr. P. Gurumurty Rao
Executive Officer
NAG Athamallik
Athamallik — 759 125, India
LIST OF PUBLICATIONS OUTSIDE THE PRESENT
THESIS WORK
It may be noted that publications from the present thesis work can be found
separately as (LIST OF PUBLICATIONS FROM THE PRESENT THESIS
WORK) in the thesis. The publications of the author out side the present thesis work
are listed below.
Peer Reviewed
1. Suresh Kumar, P., Oh, Y. M., and Cho, W. C, (2008). Surface and internal
waves scattering by partial barriers in a two-layer fluid. Journal of Korean Society
of Goastal and Ocean Engineers, 20(1), 25—33.
2. Suresh Kumar, P., (2005). Investigation of Laminar flow frictional losses in a large
hydraulic diameter pipe and annulus. Proceedings of the Institution of Mechanical
Engineers (I—Mech—E) Part C, Journal of Mechanical Engineering Science, 219(1),
53-60.
3. Salia, U. K., Malianta, P., Grinspan, A. S., Suresh Kumar, P., and Goswami, P.,
(2005). Twisted bamboo bladed rotor for Savonius wind turbines. Journal of the
Solar Energy Society of India (SESI), 4, 1 — 10.
4. Patliak, M., Suresh Kumar, P., and Salia, U. K., (2005). Prediction of off— design
performance characteristics of a gas turbine cycle using matching technique. Inter-
national Journal of Turbo and Jet Engines, 22(2), 103—119.
5. Dewan, A., Mahanta, P., Raju, K. S., and Suresh Kumar, P., (2004). Review
of passive heat transfer augmentation techniques. Proceedings of the Institution of
Mechanical Engineers (I—Mech—E) Part A, Journal of Power and Energy, 218(7),
509-527.
Conference Proceedings
1. Suresh Kumar, P., Oh, Y.M., Chan-Su, Yang., Moon-Kyung, Kang., (2008). A
study on internal waves in East-Sea near Pohang. Posture presentation in National
Workshop of Korean Society of Remote Sensing, Seoul National University, Korea.
2. Suresh Kumar, P., and Oh, Y.M., (2007). Wave trapping and energy transfer be-
tween surface and internal waves. Proceeding of Fourth Indian National Conference
on Harbor and Ocean Engineering, National Institute Of Technology Karnataka,
Surathkal, India, 811-819.
3. Suresh Kumar, P., Mahanta P., and Dewan A., (2004). Study of heat transfer
and pressure drop in a large hydraulic diameter annulus. Proceeding of Seventeenth
National Heat and Mass Transfer Conference and Sixth ISHMT/ASME Heat and
Mass Transfer Conference, Indira Gandhi Centerer for Atomic Research, Kalpakkam
India, 62—66.
4. Grinspan, A. S., Suresh Kumar, P., Saha, U. K., and Mahanta, P., (2003). Per-
formance of Savonius wind turbine rotor with twisted bamboo blades. Proceedings
of 19th Canadian Congress of Applied Mechanics, Calgary, Alberta, Canada, 2,
412-413.
5. Suresh Kumar, P., Mahanta, P., and Dewan, A., (2003). Study of laminar flow
in a large diameter annulus with twisted tape inserts. Proceedings of 2nd Interna-
tional Conference on Heat Transfer, Fluid Mechanics, and Thermodynamics, Vic-
toria Falls, Zambia, Paper No. KP3.
6. Suresh Kumar, P., Mahanta, P., and Saha, U. K., (2002). Emissions character-
istics of CNG in automobiles. Proceedings of All India Seminar on Applications of
Compressed Natural Gas (CNG) as an Automotive Fuel, Guwahati, India, 23—30.
7. Saha, U. K., Mahanta, P., Grinspan, A.S., Suresh Kumar, P., and Goswami, P.,
(2002). Vertical axis wind turbine: Design, fabrication and experimental study of
Savonius rotor made out of bamboo blades. Posture presentation in 18th National
Convention of Mechanical Engineers, National Institute of Technology, Rourkela,
India.
8. Grinspan, A. S., Suresh Kumar, P., Saha, U. K., Mahanta, P., Rao, D. V. R., and
Bhanu G. V., (2001). Design development and testing of Savonius wind turbine rotor
with twisted blades. Proceedings of 28th National Conference on Fluid Mechanics
and Fluid Power, Chandigarh, India, 428—431.
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